diff b1f9bc4..56bc728 nef_adjoint_numerical_divisors_v2.txt
diff --git a/nef_adjoint_numerical_divisors_v2.txt b/nef_adjoint_numerical_divisors_v2.txt
index dd1b709..1ae32d0 100644
--- a/nef_adjoint_numerical_divisors_v2.txt
+++ b/nef_adjoint_numerical_divisors_v2.txt
@@ -35,78 +35,78 @@ subjclass[2020]{Primary 14E30; Secondary 14C20, 14J32}
keywords{generalised abundance, numerical semiampleness, nef divisor, minimal model, Frobenius morphism}
Abstract:
-We prove the Generalised Abundance Conjecture: if $(X,B)$ is a projective
-klt $Q$-pair over an algebraically closed field of characteristic zero,
-$K_X+B$ is pseudo-effective, $M$ is a nef $Q$-Cartier divisor on $X$,
-and $K_X+B+M$ is nef, then $K_X+B+M$ is numerically equivalent to a
-semiample $Q$-Cartier divisor on $X$.
+We prove the Generalised Abundance Conjecture: if (X,B) is a projective
+klt Q-pair over an algebraically closed field of characteristic zero,
+K_X+B is pseudo-effective, M is a nef Q-Cartier divisor on X,
+and K_X+B+M is nef, then K_X+B+M is numerically equivalent to a
+semiample Q-Cartier divisor on X.
maketitle
# Introduction
-A divisor $D$ on a projective variety is nef if $Dcdot Cge 0$
-for every integral curve $C$, and semiample if some positive
-multiple $mD$ is globally generated, so that $mD$ defines a morphism to
+A divisor D on a projective variety is nef if Dcdot Cge 0
+for every integral curve C, and semiample if some positive
+multiple mD is globally generated, so that mD defines a morphism to
projective space. Nefness is numerical and semiampleness is geometric;
passing from one to the other is the central concern of the minimal
model program.
-For the log canonical divisor $K_X+B$ of a log canonical pair, the
+For the log canonical divisor K_X+B of a log canonical pair, the
relevant statement is the abundance conjecture, now a theorem in
-characteristic zero: a nef $K_X+B$ is semiample. We take it as an
-input. Adding an independent nef summand $M$ changes what can be
-expected: even if $D=K_X+B+M$ is nef, $D$ itself need not be semiample,
+characteristic zero: a nef K_X+B is semiample. We take it as an
+input. Adding an independent nef summand M changes what can be
+expected: even if D=K_X+B+M is nef, D itself need not be semiample,
and one is forced to require only a numerical semiample representative.
-A rational Cartier divisor $D$ on a projective variety $X$ is
+A rational Cartier divisor D on a projective variety X is
numerically semiample if it is numerically equivalent to a
-semiample rational Cartier divisor on the same $X$. Two rational
-Cartier divisors are numerically equivalent, written $equiv$, if their
+semiample rational Cartier divisor on the same X. Two rational
+Cartier divisors are numerically equivalent, written equiv, if their
difference meets every integral curve in zero.
[Generalised Abundance; {[cite]}]
-Let $(X,B)$ be a projective klt $Q$-pair over an algebraically closed
-field of characteristic zero. Suppose that $K_X+B$ is pseudo-effective,
-and let $M$ be a nef $Q$-Cartier divisor on $X$. If $D=K_X+B+M$ is
-nef, then there exists a semiample $Q$-Cartier divisor $L$ on $X$ with
-$Dequiv L$.
+Let (X,B) be a projective klt Q-pair over an algebraically closed
+field of characteristic zero. Suppose that K_X+B is pseudo-effective,
+and let M be a nef Q-Cartier divisor on X. If D=K_X+B+M is
+nef, then there exists a semiample Q-Cartier divisor L on X with
+Dequiv L.
-Both the boundary $B$ and the nef summand $M$ sit on $X$ itself. The
+Both the boundary B and the nef summand M sit on X itself. The
proof uses two named inputs: the log-abundance theorem of
cite[Theorem~1.1]{LA}, and the terminating minimal-model programs of
cite[Theorem~1.1 and Section~3.3]{MM}. Both are stated precisely in
Section~sec:preliminaries.
-Why numerical equivalence is necessary.: Let $E$ be a
-complex elliptic curve and let $L_alpha$ be a nontorsion line bundle
-of degree zero on $E$. Then $L_alpha$ is nef and numerically
+Why numerical equivalence is necessary.: Let E be a
+complex elliptic curve and let L_alpha be a nontorsion line bundle
+of degree zero on E. Then L_alpha is nef and numerically
trivial, but no positive multiple has a nonzero section. Since
-$K_Esim0$, taking $B=0$ and $M=L_alpha$ in Theorem~thm:main
-gives a valid hypothesis in which $K_E+M$ is nef and numerically
+K_Esim0, taking B=0 and M=L_alpha in Theorem~thm:main
+gives a valid hypothesis in which K_E+M is nef and numerically
trivial; the conclusion only asks for a semiample divisor numerically
-equivalent to $K_E+M$, which the zero divisor provides. Linear
+equivalent to K_E+M, which the zero divisor provides. Linear
equivalence would force a section, and there is none.
-The ordinary adjoint case.: For $M=0$, the conclusion of
+The ordinary adjoint case.: For M=0, the conclusion of
Theorem~thm:main is equivalent to ordinary klt abundance. One
direction is clear. For the other, Fukuda cite[Theorem~0.1]{Fukuda}
-proves that if $K_X+B$ is numerically equivalent to a semiample
-divisor, then $K_X+B$ itself is semiample. Hence Theorem~thm:main
-with $M=0$ recovers ordinary klt abundance, which is an input to the
+proves that if K_X+B is numerically equivalent to a semiample
+divisor, then K_X+B itself is semiample. Hence Theorem~thm:main
+with M=0 recovers ordinary klt abundance, which is an input to the
present proof via [cite].
-Let $(X,B)$ be a projective klt $Q$-pair over $C$ with $K_X+Bequiv 0$.
-Then every nef $Q$-Cartier divisor on $X$ is numerically semiample.
+Let (X,B) be a projective klt Q-pair over C with K_X+Bequiv 0.
+Then every nef Q-Cartier divisor on X is numerically semiample.
Proof.
-For nef $M$, the divisor $K_X+B+Mequiv M$ is nef. Apply
+For nef M, the divisor K_X+B+Mequiv M is nef. Apply
Theorem~thm:main.
@@ -115,90 +115,81 @@ step by step, to the following positivity statement about Calabi--Yau
pairs.
[Full nef dimension forces bigness]
-Let $(X,B)$ be a projective klt Calabi--Yau $Q$-pair of dimension $n$
-over $C$: that is, $K_X+Bequiv0$. Let $M$ be a nef $Q$-Cartier
-divisor on $X$ with full nef dimension, meaning every curve through a
-very general point of $X$ has positive $M$-degree. Then $M$ is big.
+Let (X,B) be a projective klt Calabi--Yau Q-pair of dimension n
+over C: that is, K_X+Bequiv0. Let M be a nef Q-Cartier
+divisor on X with full nef dimension, meaning every curve through a
+very general point of X has positive M-degree. Then M is big.
-Recall that $M$ is big if the top self-intersection $M^n$ is
-positive, equivalently if the linear systems $|kM|$ grow like $k^n$ for
-large $k$. The gap between strict positivity along very general curves
+Recall that M is big if the top self-intersection M^n is
+positive, equivalently if the linear systems |kM| grow like k^n for
+large k. The gap between strict positivity along very general curves
and positive volume is exactly where the geometry of the paper lives.
Sections~sec:jet-tools through sec:completion are devoted to
-the proof of Theorem~thm:key-ingredient for every $nge 2$.
+the proof of Theorem~thm:key-ingredient for every nge 2.
Shape of the proof.: Two simultaneous inductions on the
-dimension $n=dim X$ organise the argument.
-
-
-item[(P$_n$)] Suppose $T=K_X+B$ is pseudo-effective, $M$ is a nef
-$Q$-Cartier divisor on a projective klt $Q$-pair $(X,B)$ of dimension
-$n$, and there is a rational $a>0$ such that $(T+aM)cdot C>0$ for
-every integral curve $C$ through a very general point of $X$. Then
-$T+cM$ is big for every rational $c>0$.
-item[(Z$_n$)] Under the same hypotheses, without the
+dimension n=dim X organise the argument.
+[(P_n)] Suppose T=K_X+B is pseudo-effective, M is a nef
+Q-Cartier divisor on a projective klt Q-pair (X,B) of dimension
+n, and there is a rational a>0 such that (T+aM)cdot C>0 for
+every integral curve C through a very general point of X. Then
+T+cM is big for every rational c>0.
+[(Z_n)] Under the same hypotheses, without the
very-general-curve condition, there is a projective birational model
-$wcolon Wto X$ from a smooth projective $W$ such that the positive
-part $P_sigma(w^*(T+M))$ of the divisorial Zariski decomposition is a
-rational divisor numerically equivalent to a semiample $Q$-divisor on
-$W$.
+wcolon Wto X from a smooth projective W such that the positive
+part P_sigma(w^*(T+M)) of the divisorial Zariski decomposition is a
+rational divisor numerically equivalent to a semiample Q-divisor on
+W.
-Statement (P$_n$) produces bigness and (Z$_n$) produces numerical
-semiampleness; both are proved by induction on $n$. The case $n=0$ is
-trivial. For $n=1$: on a curve, pseudo-effective means nonnegative
-degree and nef means nonnegative degree, so (P$_n$) makes every
-positive combination ample, and (Z$_n$) makes every nef divisor
+Statement (P_n) produces bigness and (Z_n) produces numerical
+semiampleness; both are proved by induction on n. The case n=0 is
+trivial. For n=1: on a curve, pseudo-effective means nonnegative
+degree and nef means nonnegative degree, so (P_n) makes every
+positive combination ample, and (Z_n) makes every nef divisor
numerically semiample.
For the inductive step, three cases arise.
-
-
-item Positive Iitaka dimension. If $kappa(T)>0$, the
+ Positive Iitaka dimension. If kappa(T)>0, the
semiample adjoint on a suitable birational model has a
positive-dimensional base and the induction hypothesis applies to the
fibres of its contraction. This is Proposition~prop:full-positive.
-
-item Reduction to Calabi--Yau. A compatible pair of
-minimal-model programs replaces $(X,B)$ by a model $(Y,B_Y)$ on which
-the ordinary adjoint $U=K_Y+B_Y$ is semiample. If $kappa(T)=0$ then
-$Usim_Q 0$, so $(Y,B_Y)$ is a klt Calabi--Yau pair and $(P_n)$
+ Reduction to Calabi--Yau. A compatible pair of
+minimal-model programs replaces (X,B) by a model (Y,B_Y) on which
+the ordinary adjoint U=K_Y+B_Y is semiample. If kappa(T)=0 then
+Usim_Q 0, so (Y,B_Y) is a klt Calabi--Yau pair and (P_n)
reduces to Theorem~thm:key-ingredient. This is
Proposition~prop:reduction-cy.
-
-item The full-nef-dimension step. Theorem~thm:key-ingredient
+ The full-nef-dimension step. Theorem~thm:key-ingredient
is the technical core. Its proof uses two incompatible jet estimates,
produced in Section~sec:geometry and made to conflict in
Section~sec:frobenius.
Two incompatible jet estimates.: Assume, towards a
-contradiction, that $X$ is projective $Q$-factorial and terminal with
-$K_Xsim0$, and that $L$ is a nef Cartier divisor of full nef dimension
-that is not big. Since $L$ is nef and not big, $L^n=0$.
-
-Fix an ample Cartier divisor $A$ and let $tto 0$ through positive
-rationals. Let $ε>0$ be small enough that $(14ε)^n<1/4$
-and $80ε<3/4$. For suitable integers $r=r(t)to∞$ the
-divisor $P=r(L+tA)$ is ample rational Cartier with $P^ntoε^n$;
-for small $t$ the volume window
-$ε^n/2le P^nle 2ε^n$ holds. Then:
-
-
-item Small scalar bound. At a very general point $xin X$,
-every section of every $kP$ with $kP$ Cartier vanishes to order at most
-$4krho$ at $x$, where $rho=(P^n)^{1/n}$; this is
+contradiction, that X is projective Q-factorial and terminal with
+K_Xsim0, and that L is a nef Cartier divisor of full nef dimension
+that is not big. Since L is nef and not big, L^n=0.
+
+Fix an ample Cartier divisor A and let tto 0 through positive
+rationals. Let ε>0 be small enough that (14ε)^n<1/4
+and 80ε<3/4. For suitable integers r=r(t)to∞ the
+divisor P=r(L+tA) is ample rational Cartier with P^ntoε^n;
+for small t the volume window
+ε^n/2le P^nle 2ε^n holds. Then:
+ Small scalar bound. At a very general point xin X,
+every section of every kP with kP Cartier vanishes to order at most
+4krho at x, where rho=(P^n)^{1/n}; this is
Lemma~lemma:scalar-bound.
-
-item Large jet system. Over $Xtimes X$, define the
-projective bundle $Z=PP(OO_X(L)_1 ⊕ OO_X(L)_2)$ with
-tautological class $xi$, and the ample divisor $R=P_1+P_2+qxi$ for
-suitable integer $q>0$. There is a smooth point $z_*$ off the two axes
-with Seshadri constant $ε(R;z_*)>2n+2$; this is
+ Large jet system. Over Xtimes X, define the
+projective bundle Z=PP(OO_X(L)_1 ⊕ OO_X(L)_2) with
+tautological class xi, and the ample divisor R=P_1+P_2+qxi for
+suitable integer q>0. There is a smooth point z_* off the two axes
+with Seshadri constant ε(R;z_*)>2n+2; this is
Proposition~prop:large-seshadri. A Seshadri constant above
-$2n+2$ lets us generate jets at $z_*$ to that order from sections of a
-suitable Cartier multiple of $R$, via Serre vanishing and the cohomology
+2n+2 lets us generate jets at z_* to that order from sections of a
+suitable Cartier multiple of R, via Serre vanishing and the cohomology
of a point blowup.
@@ -214,59 +205,59 @@ of the kind forbidden by Lemma~lemma:bounded-curves.
The Frobenius comparison.: With the two estimates in hand,
Section~sec:frobenius derives a contradiction. A fixed finite
amount of complex data is spread out to an integral finitely generated
-$Z$-algebra of characteristic zero. For all large residual
-characteristics $p$, the two jet systems evaluate on the same
+Z-algebra of characteristic zero. For all large residual
+characteristics p, the two jet systems evaluate on the same
determinant on the diagonal of the Frobenius fiber product
-$Wtimes_{W'} W$. The scalar bound makes that determinant vanish at
-the diagonal to order at most $80ε+O(1/p)$ in the relevant
+Wtimes_{W'} W. The scalar bound makes that determinant vanish at
+the diagonal to order at most 80ε+O(1/p) in the relevant
normalization, while the diagonal-rank bound from the Frobenius
-filtration makes the same order of vanishing at least $3/4$. Since
-$80ε<3/4$ and $O(1/p)to 0$, the two bounds are incompatible for
-all large $p$; this is Theorem~thm:finite-data, whose proof
+filtration makes the same order of vanishing at least 3/4. Since
+80ε<3/4 and O(1/p)to 0, the two bounds are incompatible for
+all large p; this is Theorem~thm:finite-data, whose proof
occupies Section~sec:frobenius.
Extension to klt Calabi--Yau pairs.: Once
-Theorem~thm:key-ingredient is proved for terminal $X$ with trivial
+Theorem~thm:key-ingredient is proved for terminal X with trivial
boundary, Section~sec:completion extends it in two steps. First,
-a klt $X$ with $K_Xequiv 0$ admits a finite quasi-'etale index-one
-cover $picolon X^{ind}to X$ with $K_{X^{ind}}sim 0$
+a klt X with K_Xequiv 0 admits a finite quasi-'etale index-one
+cover picolon X^{ind}to X with K_{X^{ind}}sim 0
and canonical singularities; a crepant terminalization then gives a
-terminal $Q$-factorial $widetilde Xto X^{ind}$. Full nef
+terminal Q-factorial widetilde Xto X^{ind}. Full nef
dimension and bigness both survive the finite quasi-'etale cover, so
-the terminal trivial-boundary case gives bigness of $M$ on $X$. Second,
-for an arbitrary effective boundary $B$ with $K_X+Bequiv 0$, a small
-perturbation $(X,(1+alpha)B)$ together with a pair of MMPs either
-reduces to the positive-Iitaka case or forces $B_Y=0$; in the latter
-case $K_Ysim_Q0$ and $M$ remains nef of full nef dimension, and the
-terminal trivial-boundary case makes $M$ big.
+the terminal trivial-boundary case gives bigness of M on X. Second,
+for an arbitrary effective boundary B with K_X+Bequiv 0, a small
+perturbation (X,(1+alpha)B) together with a pair of MMPs either
+reduces to the positive-Iitaka case or forces B_Y=0; in the latter
+case K_Ysim_Q0 and M remains nef of full nef dimension, and the
+terminal trivial-boundary case makes M big.
Positive parts and numerical semiampleness.: Once both
-inductions hold, Section~sec:positive proves (Z$_n$). First, a
-large-nef-adjoint MMP trivial on a multiple of $M$ may be arranged so
-that $T+mL$ becomes nef. On the resulting model either $T+mL$ has full
-nef dimension, in which case (P$_n$) gives bigness of $T+M$ and
+inductions hold, Section~sec:positive proves (Z_n). First, a
+large-nef-adjoint MMP trivial on a multiple of M may be arranged so
+that T+mL becomes nef. On the resulting model either T+mL has full
+nef dimension, in which case (P_n) gives bigness of T+M and
Kodaira's lemma exhibits a semiample positive part; or the nef
-reduction of $T+mL$ is nontrivial, in which case $M$ descends to a
-lower-dimensional base variety $Z_2$, the induction applies to the
+reduction of T+mL is nontrivial, in which case M descends to a
+lower-dimensional base variety Z_2, the induction applies to the
base pair, and the resulting positive part pulls back along the
fibration.
Finally, the positive part on a smooth model descends to the original
variety, because a klt variety has rational singularities so
-$Pic^0$ matches along the pullback of any birational model. A
-finite-type spread argument completes the passage from $C$ to any
+Pic^0 matches along the pullback of any birational model. A
+finite-type spread argument completes the passage from C to any
algebraically closed field of characteristic zero.
Organization.: Section~sec:preliminaries records
conventions and the two input theorems. Section~sec:reduction
-sets up the two inductions and reduces (P$_n$) to the Calabi--Yau
+sets up the two inductions and reduces (P_n) to the Calabi--Yau
case. Section~sec:jet-tools proves the bounded-degree
obstruction and records the moving-kernel tools from [cite].
Sections~sec:geometry and~sec:frobenius prove the terminal
Calabi--Yau case of Theorem~thm:key-ingredient, and
Section~sec:completion extends it to all klt Calabi--Yau pairs.
Section~sec:positive completes the inductions and descends to
-$X$; Section~sec:field handles general algebraically closed
+X; Section~sec:field handles general algebraically closed
fields.
Relation to earlier work.: Lazi'c--Peternell [cite]
@@ -274,7 +265,7 @@ formulated generalised abundance as the numerical extension of
abundance and established it on a Calabi--Yau variety under a
Semiampleness Conjecture plus termination and abundance in lower
dimensions. Their Corollaries C and D settle surfaces and threefolds
-with $kappa(K_X+B)>0$ unconditionally. Subsequent work by
+with kappa(K_X+B)>0 unconditionally. Subsequent work by
Lazi'c--Peternell [cite], Chaudhuri, and Fontanari addressed
numerical nonvanishing, small-dimensional nef reductions, and
semiampleness for surfaces. The strategy here follows
@@ -289,33 +280,33 @@ bounded-degree curves for the moving-jet machinery.
# Conventions and standing inputs
-Throughout, unless stated otherwise, the base field is $C$. Varieties
+Throughout, unless stated otherwise, the base field is C. Varieties
are integral and projective. Grading of local inputs is stated at
each appearance.
## Log pairs and singularities
-A $Q$-pair $(X,B)$ consists of a normal projective variety $X$,
-an effective rational boundary $B$, and a $Q$-Cartier divisor $K_X+B$.
-Log discrepancies are defined on a log resolution $hcolon Yto X$,
-via the equality $K_Y+B_Y=h^*(K_X+B)$: a prime divisor $Esubset Y$
-with coefficient $b_E$ in $B_Y$ carries log discrepancy $1-b_E$.
+A Q-pair (X,B) consists of a normal projective variety X,
+an effective rational boundary B, and a Q-Cartier divisor K_X+B.
+Log discrepancies are defined on a log resolution hcolon Yto X,
+via the equality K_Y+B_Y=h^*(K_X+B): a prime divisor Esubset Y
+with coefficient b_E in B_Y carries log discrepancy 1-b_E.
The pair is klt if all log discrepancies are positive and
log canonical if they are nonnegative. Standard cone,
contraction, negativity, and resolution theorems are used as in
[cite].
-The symbols $sim_Q$ and $equiv$ denote rational linear and numerical
-equivalence. A divisor $D$ is pseudo-effective if its class in
-$N^1(X)_R$ lies in the closure of the cone generated by effective
+The symbols sim_Q and equiv denote rational linear and numerical
+equivalence. A divisor D is pseudo-effective if its class in
+N^1(X)_R lies in the closure of the cone generated by effective
divisors; big if its class lies in the interior of that cone;
-nef if $Dcdot Cge0$ for every integral curve $C$. A divisor
+nef if Dcdot Cge0 for every integral curve C. A divisor
is semiample if some positive multiple is globally generated.
-The Iitaka dimension $kappa(D)$ is the maximal dimension of
-the image of a map defined by a linear system $|kD|$, for $k$ verifying
-$h^0(kD)>0$.
+The Iitaka dimension kappa(D) is the maximal dimension of
+the image of a map defined by a linear system |kD|, for k verifying
+h^0(kD)>0.
-Over $C$, a very general point is a point outside a specified
+Over C, a very general point is a point outside a specified
countable union of proper closed subsets. Most statements below
specify the countable union only implicitly; iterated applications
remain valid because a countable union of countable unions remains
@@ -323,21 +314,19 @@ countable.
## Nef reduction
-For a nef divisor $N$ on a projective variety, the nef
+For a nef divisor N on a projective variety, the nef
reduction [cite] is an almost holomorphic dominant
-rational map $picolon Xdashrightarrow Z$ with connected compact
+rational map picolon Xdashrightarrow Z with connected compact
general fibres such that:
+ N|_Fequiv0 on a very general compact fibre F;
+ if C is a curve through a very general point of X not
+contracted by pi, then Ncdot C>0.
-item $N|_Fequiv0$ on a very general compact fibre $F$;
-item if $C$ is a curve through a very general point of $X$ not
-contracted by $pi$, then $Ncdot C>0$.
-
-
-The base dimension $dim Z$ is the nef dimension $n(X,N)$. It
-depends only on the numerical class of $N$. By construction,
-$n(X,N)=dim X$ if and only if every curve through a very general
-point has positive $N$-degree. This is the condition called
+The base dimension dim Z is the nef dimension n(X,N). It
+depends only on the numerical class of N. By construction,
+n(X,N)=dim X if and only if every curve through a very general
+point has positive N-degree. This is the condition called
full nef dimension. Crucially, full nef dimension is purely
numerical: it does not assert or imply bigness. The gap between full
nef dimension and bigness is exactly the gap that
@@ -345,44 +334,44 @@ Theorem~thm:key-ingredient closes for Calabi--Yau pairs.
## Divisorial Zariski decomposition
-For a pseudo-effective real divisor $D$ on a smooth projective
-variety, and a prime divisor $Gamma$ on it, fix an ample divisor $A$
+For a pseudo-effective real divisor D on a smooth projective
+variety, and a prime divisor Gamma on it, fix an ample divisor A
and put
[
sigma_Gamma(D)=lim_{delta↓0}
inf{mult_Gamma D':D'ge0, D'sim_R D+delta A}.
]
These asymptotic divisorial orders are finite, nonnegative, and
-independent of $A$. Only finitely many of them are nonzero, and each
-depends only on the numerical class of $D$. The
+independent of A. Only finitely many of them are nonzero, and each
+depends only on the numerical class of D. The
divisorial Zariski decomposition is
[
N_sigma(D)=sum_Gammasigma_Gamma(D)Gamma,
- qquad P_sigma(D)=D-N_sigma(D).
+ P_sigma(D)=D-N_sigma(D).
]
-We call $N_sigma(D)$ the negative part and $P_sigma(D)$ the
-positive part. The sum is a finite $R$-linear combination of
-prime divisors; the difference $P_sigma(D)$ is a pseudo-effective
-$R$-divisor whose negative part is trivial. A divisor $D$ is nef if
-and only if $N_sigma(D)=0$. Otherwise, the negative part captures
-exactly the ``asymptotically fixed'' divisorial component of $D$.
+We call N_sigma(D) the negative part and P_sigma(D) the
+positive part. The sum is a finite R-linear combination of
+prime divisors; the difference P_sigma(D) is a pseudo-effective
+R-divisor whose negative part is trivial. A divisor D is nef if
+and only if N_sigma(D)=0. Otherwise, the negative part captures
+exactly the ``asymptotically fixed'' divisorial component of D.
## The very general point lemma
-Let $D$ be a pseudo-effective rational Cartier divisor on a projective
-variety $X$ over $C$. Then every integral curve through a
-sufficiently very general point of $X$ has nonnegative $D$-degree.
+Let D be a pseudo-effective rational Cartier divisor on a projective
+variety X over C. Then every integral curve through a
+sufficiently very general point of X has nonnegative D-degree.
Proof.
-Fix an ample Cartier divisor $A$ and rationals $delta_i>0$ with
-$delta_ito 0$. Each $D+delta_iA$ is big, so there is an effective
-rational divisor $E_isim_Q D+delta_iA$. A point outside the
-countable union $∪_iSupp E_i$ is very general, and every curve
-$C$ through such a point meets no $Supp E_i$. Therefore
-$(D+delta_iA)cdot Cge0$ for every $i$, and $Dcdot Cge0$ after
-letting $ito∞$.
+Fix an ample Cartier divisor A and rationals delta_i>0 with
+delta_ito 0. Each D+delta_iA is big, so there is an effective
+rational divisor E_isim_Q D+delta_iA. A point outside the
+countable union ∪_iSupp E_i is very general, and every curve
+C through such a point meets no Supp E_i. Therefore
+(D+delta_iA)cdot Cge0 for every i, and Dcdot Cge0 after
+letting ito∞.
## Two input theorems
@@ -392,7 +381,7 @@ zero; both are used as black boxes.
The first input is ordinary log abundance. Recall that a klt variety
has rational singularities, so pullbacks from an arbitrary birational
-model do not affect $Pic^0$; the distinction between a rational
+model do not affect Pic^0; the distinction between a rational
divisor and the line bundle it defines is thus controlled on smooth
models (see Lemma~lemma:numsemiample-descent). The log
abundance statement asserts that this subtlety disappears entirely
@@ -400,38 +389,38 @@ for nef log canonical adjoints.
[Log abundance, {cite[Theorem~1.1]{LA}}]
-Let $(X,B)$ be a projective log canonical $Q$-pair over an
-algebraically closed field of characteristic zero. If $K_X+B$ is
+Let (X,B) be a projective log canonical Q-pair over an
+algebraically closed field of characteristic zero. If K_X+B is
nef, then it is semiample.
The second input provides the minimal-model programs used in
Section~sec:reduction, both for ordinary adjoints and for
generalized pairs whose only polarized data is a nef
-$Q$-Cartier $b$-divisor. The statement records a terminating
+Q-Cartier b-divisor. The statement records a terminating
choice of program: at each step there are many possible
-$(K_X+B+M)$-negative extremal rays, but the theorem gives a
+(K_X+B+M)-negative extremal rays, but the theorem gives a
fixed order in which they have been contracted that terminates after
finitely many steps.
A nef rational Cartier b-divisor on a normal projective variety
-$X$ is specified by a nef rational Cartier divisor $N'$ on a projective
-birational model $Yto X$; throughout the model $Y$, and on every
-higher birational model, $N'$ pulls back unchanged. Generalized
+X is specified by a nef rational Cartier divisor N' on a projective
+birational model Yto X; throughout the model Y, and on every
+higher birational model, N' pulls back unchanged. Generalized
discrepancies are computed by pulling back the generalized adjoint
-while keeping $N'$ fixed on all higher models; the generalized pair is
+while keeping N' fixed on all higher models; the generalized pair is
generalized klt when all generalized discrepancies are positive. The
-trace of $M$ on a lower model is its pushforward, which need
+trace of M on a lower model is its pushforward, which need
not be nef.
[Minimal-model programs,
{cite[Theorem~1.1 and Section~3.3]{MM}}]
-Let $(X,B+mathbf M)$bfM)$ be a projective generalized klt $Q$-pair over
-$C$, with $X$ $Q$-factorial and with fixed nef rational Cartier
-b-divisor data $mathbf M$.bfM$. If $K_X+B+M_X$ is pseudo-effective, there
+Let (X,B+mathbf M)bfM) be a projective generalized klt Q-pair over
+C, with X Q-factorial and with fixed nef rational Cartier
+b-divisor data mathbf M.bfM. If K_X+B+M_X is pseudo-effective, there
is a terminating choice of its minimal-model program, with
-$Q$-factorial generalized klt models and a nef endpoint. The program
+Q-factorial generalized klt models and a nef endpoint. The program
extracts no divisors and generalized log discrepancies do not decrease.
On a common resolution of its initial and final models, the pullback of
the initial adjoint is the pullback of the final adjoint plus an
@@ -439,7 +428,7 @@ effective divisor exceptional over the final model.
Two remarks. First, the ordinary klt case of
-Theorem~input:models arises by taking $mathbf M=0$;bfM=0$; then
+Theorem~input:models arises by taking mathbf M=0;bfM=0; then
Theorem~input:abundance makes the endpoint a good minimal model.
Second, neither theorem above asserts abundance for a nonzero
generalized nef summand. Their combination still exposes the
@@ -450,123 +439,122 @@ Theorem~thm:key-ingredient.
# The two inductions and reduction to Calabi--Yau
-We work over $C$ in this section. The reductions follow the strategy
+We work over C in this section. The reductions follow the strategy
of Lazi'c--Peternell cite[Sections~4--5]{LP}, with the terms made
explicit. The essential point is to retain control of two adjoints at
-once: the ordinary adjoint $T=K_X+B$, whose semiampleness is supplied
-by abundance, and a nearby generalized nef adjoint $T+sM$ whose
+once: the ordinary adjoint T=K_X+B, whose semiampleness is supplied
+by abundance, and a nearby generalized nef adjoint T+sM whose
well-behaved full nef dimension lets bigness descend from a
Calabi--Yau model. The two adjoints live on the same birational model
and must remain tractable simultaneously.
## The two induction statements
-For a projective klt $Q$-pair $(X,B)$ and a nef $Q$-Cartier divisor
-$M$ on $X$, write $T=K_X+B$. Assume $T$ is pseudo-effective, and
-consider the following condition: for some $ainQ_{>0}$,
+For a projective klt Q-pair (X,B) and a nef Q-Cartier divisor
+M on X, write T=K_X+B. Assume T is pseudo-effective, and
+consider the following condition: for some ainQ_{>0},
(T+aM)cdot C>0
- quadfor every integral curve $C$ through a very general point of $X$.
+ quadfor every integral curve C through a very general point of X.
-Note that $T+aM$ need not be nef; the condition is on the class it
+Note that T+aM need not be nef; the condition is on the class it
defines, and only at the very general locus.
Lemma~lemma:pseudo-vg implies that
-(eq:full-condition) holds whenever $M$ has full nef dimension.
-If $T+aM$ is nef, then (eq:full-condition) is exactly the
-assertion that $T+aM$ has full nef dimension.
+(eq:full-condition) holds whenever M has full nef dimension.
+If T+aM is nef, then (eq:full-condition) is exactly the
+assertion that T+aM has full nef dimension.
We prove the following two statements simultaneously by induction on
-$n=dim X$.
-
-item[(P$_n$)] Let $(X,B)$ be a projective klt $Q$-pair of dimension
-$n$, let $M$ be a nef $Q$-Cartier divisor on $X$, let $T=K_X+B$ be
+n=dim X.
+[(P_n)] Let (X,B) be a projective klt Q-pair of dimension
+n, let M be a nef Q-Cartier divisor on X, let T=K_X+B be
pseudo-effective, and suppose (eq:full-condition) holds for some
-rational $a>0$. Then $T+cM$ is big for every rational $c>0$.
-item[(Z$_n$)] Under the same hypotheses on $(X,B)$, $M$, $T$ but
+rational a>0. Then T+cM is big for every rational c>0.
+[(Z_n)] Under the same hypotheses on (X,B), M, T but
without (eq:full-condition), there is a projective birational
-morphism $wcolon Wto X$ from a smooth projective variety such that
-$P_sigma(w^*(T+M))$ is rational and numerically equivalent to a
-semiample $Q$-divisor on $W$.
+morphism wcolon Wto X from a smooth projective variety such that
+P_sigma(w^*(T+M)) is rational and numerically equivalent to a
+semiample Q-divisor on W.
-Both statements hold in dimensions $0$ and $1$. For $n=0$ there is
-nothing to prove. For $n=1$, pseudo-effectivity of $T$ and nefness of
-$M$ both say their degrees are nonnegative; then
+Both statements hold in dimensions 0 and 1. For n=0 there is
+nothing to prove. For n=1, pseudo-effectivity of T and nefness of
+M both say their degrees are nonnegative; then
(eq:full-condition) makes every positive combination have
-positive degree, hence ample, and (P$_1$) is immediate. Similarly a
+positive degree, hence ample, and (P_1) is immediate. Similarly a
nef divisor on a curve has zero negative part in the divisorial
Zariski decomposition, so it is itself semiample in its own numerical
-class; this gives (Z$_1$).
+class; this gives (Z_1).
-Fix $nge2$, and assume (P$_j$) and (Z$_j$) for every $j<n$. We
-verify (P$_n$) and (Z$_n$). Along the way we may freely replace $X$
-by a small projective $Q$-factorialization (which pulls back the
-adjoint crepantly and the nef divisor $M$ to a nef divisor), and
-replace a statement about $X$ by the corresponding statement about a
+Fix nge2, and assume (P_j) and (Z_j) for every j<n. We
+verify (P_n) and (Z_n). Along the way we may freely replace X
+by a small projective Q-factorialization (which pulls back the
+adjoint crepantly and the nef divisor M to a nef divisor), and
+replace a statement about X by the corresponding statement about a
smooth birational model. Indeed (eq:full-condition) persists
to the isomorphism locus, bigness descends under the birational
-pullback, and a smooth model on which $P_sigma$ is rational and
-numerically semiample descends to $X$ by
+pullback, and a smooth model on which P_sigma is rational and
+numerically semiample descends to X by
Lemma~lemma:numsemiample-descent. So we are free to impose
-$Q$-factoriality when convenient.
+Q-factoriality when convenient.
## The elementary ray bound for generalized pairs
We start with the only new cone-theoretic input: a uniformity statement
for negative extremal rays of generalized klt pairs. Recall that a
-generalized klt pair $(V,Delta+mathbf N)$bfN)$ has boundary $Delta$ and
+generalized klt pair (V,Delta+mathbf N)bfN) has boundary Delta and
polarized data given by a fixed nef rational Cartier b-divisor
-$mathbf N$bfN$ determined on some projective birational model of $V$.
-The notation $N_V$ denotes the trace of $mathbf N$bfN$ on $V$; this trace
+mathbf NbfN determined on some projective birational model of V.
+The notation N_V denotes the trace of mathbf NbfN on V; this trace
need not be nef, only the polarized data on the fixing model is.
-Let $(V,Delta+mathbf N)$bfN)$ be a projective generalized klt $Q$-pair
-of dimension $n$, with effective boundary, $Q$-factorial $V$, and
+Let (V,Delta+mathbf N)bfN) be a projective generalized klt Q-pair
+of dimension n, with effective boundary, Q-factorial V, and
nef rational Cartier b-divisor data. Every
-$(K_V+Delta+N_V)$-negative extremal ray has an integral rational curve
-generator $C$ with
+(K_V+Delta+N_V)-negative extremal ray has an integral rational curve
+generator C with
[
-4nle (K_V+Delta+N_V)cdot C < 0.
]
The contraction, and the flip when it is small, can be realized using
an ordinary klt adjoint negative on the same ray. Birational
-minimal-model outputs remain $Q$-factorial. On a common resolution
+minimal-model outputs remain Q-factorial. On a common resolution
of a birational step, the pullback of the input generalized adjoint
equals the pullback of its output transform plus an effective divisor
exceptional over the output.
-The constant $4n$ is twice the usual bound $2n$ for an ordinary klt
-pair. The factor $2$ arises because the argument passes through a
-small perturbation of $G=K_V+Delta+N_V$ by an ample divisor to reach
+The constant 4n is twice the usual bound 2n for an ordinary klt
+pair. The factor 2 arises because the argument passes through a
+small perturbation of G=K_V+Delta+N_V by an ample divisor to reach
a genuine klt adjoint; the perturbation inflates the negativity of
-$G$ along the chosen ray by at most a factor of two.
+G along the chosen ray by at most a factor of two.
Proof.
-Set $G=K_V+Delta+N_V$ and fix a $G$-negative extremal ray generated
-by $R$. Choose an ample rational divisor $A$ on $V$ and a rational
-$eta>0$ small enough that $G+eta A$ is still negative on $R$, and
+Set G=K_V+Delta+N_V and fix a G-negative extremal ray generated
+by R. Choose an ample rational divisor A on V and a rational
+eta>0 small enough that G+eta A is still negative on R, and
such that
-$-(G+eta A)cdotgammage-tfrac12 Gcdotgamma$ for every nonzero
-class $gamma$ on $R$.
+-(G+eta A)cdotgammage-tfrac12 Gcdotgamma for every nonzero
+class gamma on R.
-We need to represent $G+eta A$ as an ordinary klt adjoint. Because
-the b-divisor $mathbf N$bfN$ may not descend to $V$, work upstairs.
-Choose a projective log resolution $hcolon V'to V$ carrying
-$mathbf N$bfN$ as a locally free sheaf, and write
+We need to represent G+eta A as an ordinary klt adjoint. Because
+the b-divisor mathbf NbfN may not descend to V, work upstairs.
+Choose a projective log resolution hcolon V'to V carrying
+mathbf NbfN as a locally free sheaf, and write
[
K_{V'}+Delta'+N'=h^*G
]
-with coefficients of $Delta'$ strictly less than one; this is
-generalized klt. Choose $V'$ with an effective exceptional $F$ such
-that $-F$ is $h$-ample. For small $ε>0$ the divisor
-$N'+eta h^*A-ε F$ is ample (being the sum of an ample and a
-nef, adjusted by an $h$-ample), and a general effective rational
+with coefficients of Delta' strictly less than one; this is
+generalized klt. Choose V' with an effective exceptional F such
+that -F is h-ample. For small ε>0 the divisor
+N'+eta h^*A-ε F is ample (being the sum of an ample and a
+nef, adjusted by an h-ample), and a general effective rational
representative of this ample class can be chosen with coefficients
-small enough that, after adding $ε F$, it preserves the sub-klt
-condition on $Delta'$. Pushing that representative to $V$ gives an
-effective rational divisor $Θ$ on $V$ with
+small enough that, after adding ε F, it preserves the sub-klt
+condition on Delta'. Pushing that representative to V gives an
+effective rational divisor Θ on V with
[
K_V+Θ sim_Q G+eta A.
]
@@ -574,25 +562,25 @@ The crepant equality is checked upstairs: its difference is
exceptional and relatively numerically trivial, hence zero by the
negativity lemma.
-The ordinary cone theorem for the klt pair $(V,Θ)$ gives a
-rational curve generator $C$ of the ray with
+The ordinary cone theorem for the klt pair (V,Θ) gives a
+rational curve generator C of the ray with
[
0<-(G+eta A)cdot Cle2n.
]
The chain
-$-(G+eta A)cdot Cge-tfrac12 Gcdot C$
-gives $-Gcdot Cle 4n$.
-The contraction $ccolon Vto S$ of the ray for $G$ is the contraction
-of the same ray for the ordinary klt pair $(V,Θ)$, by
+-(G+eta A)cdot Cge-tfrac12 Gcdot C
+gives -Gcdot Cle 4n.
+The contraction ccolon Vto S of the ray for G is the contraction
+of the same ray for the ordinary klt pair (V,Θ), by
cone-contraction uniqueness, and the flip behaves identically: since
-$Pic(V/S)$ is one-dimensional, there is a rational $lambda>0$ with
-$G-lambda(K_V+Θ)$ numerically trivial over $S$, and Cartier
-descent for the ordinary pair gives a pullback divisor from $S$ in
-the appropriate class; on the flipped model $V^+$, the same
-observation shows $G^+$ is ample over $S$. The $Q$-factoriality of
-$V^+$ then follows from the ordinary flip, and the effective
-exceptional comparison of $G$ is the sum of its ordinary counterpart
-with $lambda$, since the $S$-pullback parts cancel.
+Pic(V/S) is one-dimensional, there is a rational lambda>0 with
+G-lambda(K_V+Θ) numerically trivial over S, and Cartier
+descent for the ordinary pair gives a pullback divisor from S in
+the appropriate class; on the flipped model V^+, the same
+observation shows G^+ is ample over S. The Q-factoriality of
+V^+ then follows from the ordinary flip, and the effective
+exceptional comparison of G is the sum of its ordinary counterpart
+with lambda, since the S-pullback parts cancel.
This proves the lemma.
@@ -605,27 +593,25 @@ on a lower-dimensional base, while a small multiple of the nef summand
stays tractable on the same model.
-Let $(X,B)$ be a projective $Q$-factorial klt $Q$-pair of dimension
-$n$, let $T=K_X+B$ be pseudo-effective, and let $M$ be a nef
-$Q$-Cartier divisor on $X$. Suppose (eq:full-condition) holds.
-Then there are a birational contraction $Xdashrightarrow Y$, a number
-$s_0inQ_{>0}$, and a common smooth projective resolution
+Let (X,B) be a projective Q-factorial klt Q-pair of dimension
+n, let T=K_X+B be pseudo-effective, and let M be a nef
+Q-Cartier divisor on X. Suppose (eq:full-condition) holds.
+Then there are a birational contraction Xdashrightarrow Y, a number
+s_0inQ_{>0}, and a common smooth projective resolution
[
Xxleftarrow{p}Wxrightarrow{q}Y
]
-with the following properties. If $B_Y$, $N$ denote the transforms of
-$B$ and $M$, and $U=K_Y+B_Y$, then:
-
-
-item $(Y,B_Y)$ is klt and $Q$-factorial, $U$ is semiample, and
-$kappa(Y,U)=kappa(X,T)ge0$.
-item $N$ is pseudo-effective, and $U+s_0N$ is nef of full nef
+with the following properties. If B_Y, N denote the transforms of
+B and M, and U=K_Y+B_Y, then:
+ (Y,B_Y) is klt and Q-factorial, U is semiample, and
+kappa(Y,U)=kappa(X,T)ge0.
+ N is pseudo-effective, and U+s_0N is nef of full nef
dimension.
-item There are effective $q$-exceptional rational divisors $E_0,E_1$
-on $W$ with
+ There are effective q-exceptional rational divisors E_0,E_1
+on W with
p^*T=q^*U+E_0,
- qquad
+
p^*(T+s_0M)=q^*(U+s_0N)+E_1.
@@ -633,117 +619,117 @@ on $W$ with
The two comparisons of (eq:two-comparisons) are the effective
exceptional adjoint comparisons of the two minimal-model programs; the
-constant $s_0$ is chosen so that the second program's rays are already
-negative for the first program's adjoint at coefficient $s$, and $s_0$
+constant s_0 is chosen so that the second program's rays are already
+negative for the first program's adjoint at coefficient s, and s_0
is small enough that a proportionality argument with a free Cartier
-multiple $dU_0$ rules out any positivity of $U_0$ along such a ray.
+multiple dU_0 rules out any positivity of U_0 along such a ray.
Proof.
-Apply Theorem~input:models to the ordinary klt pair $(X,B)$
-(i.e. with $mathbf M=0$)bfM=0$) and pseudo-effective adjoint $T$. The
-output is a terminating $T$-MMP ending on a $Q$-factorial klt pair
-$(Y_0,B_0)$ with nef ordinary adjoint $U_0=K_{Y_0}+B_0$.
-Theorem~input:abundance makes $U_0$ semiample. The
+Apply Theorem~input:models to the ordinary klt pair (X,B)
+(i.e. with mathbf M=0)bfM=0) and pseudo-effective adjoint T. The
+output is a terminating T-MMP ending on a Q-factorial klt pair
+(Y_0,B_0) with nef ordinary adjoint U_0=K_{Y_0}+B_0.
+Theorem~input:abundance makes U_0 semiample. The
comparisons along the finitely many birational steps give
-$kappa(X,T)=kappa(Y_0,U_0)ge0$.
+kappa(X,T)=kappa(Y_0,U_0)ge0.
-Let $N_0$ be the transform of $M$ on $Y_0$. Because the first
-program has finitely many steps, there is a rational $s>0$ such that
-every step is still negative for the transform of $T+uM$ for every
-$uin(0,s]$, and such that the strict sign on the flipped sides is
+Let N_0 be the transform of M on Y_0. Because the first
+program has finitely many steps, there is a rational s>0 such that
+every step is still negative for the transform of T+uM for every
+uin(0,s], and such that the strict sign on the flipped sides is
preserved. Hence these steps are generalized klt steps for the pairs
-$(X,B+umathbf M)$,bfM)$, so $(Y_0,B_0+umathbf M)$bfM)$ is a generalized klt
-pair for $uin(0,s]$. In particular its trace
-$(Y_0,B_0+smathbf M)$bfM)$ is a generalized klt pair with polarized data
-$smathbf M$bfM$ contained in the fixed b-divisor. Pushforward under a
-birational contraction to a $Q$-factorial target preserves
+(X,B+umathbf M),bfM), so (Y_0,B_0+umathbf M)bfM) is a generalized klt
+pair for uin(0,s]. In particular its trace
+(Y_0,B_0+smathbf M)bfM) is a generalized klt pair with polarized data
+smathbf MbfM contained in the fixed b-divisor. Pushforward under a
+birational contraction to a Q-factorial target preserves
pseudo-effectivity: on a common resolution, an upstairs divisor
numerically trivial implies it is the pullback of its pushforward,
thanks to the negativity lemma, and effective approximation then
-proves the claim for classes. Thus $N_0$ is pseudo-effective.
+proves the claim for classes. Thus N_0 is pseudo-effective.
-Choose a positive integer $d$ such that $dU_0$ is Cartier and
-globally generated. Pick $s_0in(0,s)$ rational with
+Choose a positive integer d such that dU_0 is Cartier and
+globally generated. Pick s_0in(0,s) rational with
(1-s_0/s)/(d)>4n(s_0)/(s).
-The generalized adjoint $U_0+s_0N_0$ is pseudo-effective.
+The generalized adjoint U_0+s_0N_0 is pseudo-effective.
Theorem~input:models gives a terminating program for this
generalized adjoint. We claim that every step of this program is
-$U_0$-trivial and is also negative for the generalized adjoint with
-coefficient $s$, where throughout the claim the symbols denote their
+U_0-trivial and is also negative for the generalized adjoint with
+coefficient s, where throughout the claim the symbols denote their
transforms.
-Suppose inductively that $U_i$ is Cartier and free, and that the
-generalized pair with coefficient $s$ is generalized klt. For a
-$(U_i+s_0N_i)$-negative ray, the decomposition
+Suppose inductively that U_i is Cartier and free, and that the
+generalized pair with coefficient s is generalized klt. For a
+(U_i+s_0N_i)-negative ray, the decomposition
[
U_i+s_0N_i=(1-s_0/s)U_i+(s_0/s)(U_i+sN_i)
]
-together with nefness of $U_i$ shows the ray is
-$(U_i+sN_i)$-negative. Pick its rational curve generator $C$ from
-Lemma~lemma:generalized-ray-bound. If $U_icdot C>0$,
-integrality of the free $dU_i$ gives $U_icdot Cge1/d$, so
+together with nefness of U_i shows the ray is
+(U_i+sN_i)-negative. Pick its rational curve generator C from
+Lemma~lemma:generalized-ray-bound. If U_icdot C>0,
+integrality of the free dU_i gives U_icdot Cge1/d, so
[
(U_i+s_0N_i)cdot C
ge(1-s_0/s)/(d)-4n(s_0)/(s)>0
]
by (eq:small-nef-coefficient); contradiction. Thus
-$U_icdot C=0$.
+U_icdot C=0.
-The morphism defined by the free $dU_i$ is constant on each fibre of
+The morphism defined by the free dU_i is constant on each fibre of
the extremal contraction, so it descends to the base; pulling back to
-the flipped model gives the transform of $dU_i$, again Cartier and
-free. Negativity for the generalized adjoint with coefficient $s$
-persists, so the generalized klt condition on the pair $(B_i+smathbf
-M)$bfM)$ continues to hold. The induction closes on the finite program.
-
-Let $Y$ be its output. Because each step is $U_0$-trivial,
-$U=U_0$ is pulled back unchanged, hence semiample with the same
-Iitaka dimension $kappa(X,T)$. The transform of the generalized
-adjoint is $U+s_0N$, which is nef by the construction of the program.
+the flipped model gives the transform of dU_i, again Cartier and
+free. Negativity for the generalized adjoint with coefficient s
+persists, so the generalized klt condition on the pair (B_i+smathbf
+M)bfM) continues to hold. The induction closes on the finite program.
+
+Let Y be its output. Because each step is U_0-trivial,
+U=U_0 is pulled back unchanged, hence semiample with the same
+Iitaka dimension kappa(X,T). The transform of the generalized
+adjoint is U+s_0N, which is nef by the construction of the program.
Combining the pullback comparisons of the two programs gives
(eq:two-comparisons).
-It remains to prove that $U+s_0N$ has full nef dimension. Suppose
+It remains to prove that U+s_0N has full nef dimension. Suppose
toward a contradiction that its nef reduction has a positive-dimensional
-very general compact fibre $F$. Shrink the base so that these compact
+very general compact fibre F. Shrink the base so that these compact
fibres lie inside the morphism locus of the almost holomorphic map.
-The image in $Y$ of the exceptional locus of $q$ has codimension
-$ge2$; a very general fibre $F$ therefore misses it in codimension
-$ge2$ at every component, by the fibre dimension theorem applied to
-the components that dominate the base. We can pick a curve $Csubset
-F$ passing through a point chosen with the very general properties of
+The image in Y of the exceptional locus of q has codimension
+ge2; a very general fibre F therefore misses it in codimension
+ge2 at every component, by the fibre dimension theorem applied to
+the components that dominate the base. We can pick a curve Csubset
+F passing through a point chosen with the very general properties of
(eq:full-condition): outside the exceptional image, and in the
-common isomorphism locus of the first program. If $dim F>1$, take
-$C$ to be a sufficiently general complete intersection in $F$
-through that point; if $dim F=1$, take $C=F$ itself.
-
-Because $U+s_0N$ is nef with positive-dimensional nef reduction and
-$U$ is semiample, both $U$ and $N$ have degree zero on $C$: this
-follows from Lemma~lemma:pseudo-vg applied to each, since $U$ is
-nef and $N$ pseudo-effective and their positive combination gives a
-nonnegative, hence zero, degree along $F$. The strict lift
-$widetilde Csubset W$ avoids the effective exceptional divisors
-$E_0,E_1$. Both equalities in (eq:two-comparisons) then give
+common isomorphism locus of the first program. If dim F>1, take
+C to be a sufficiently general complete intersection in F
+through that point; if dim F=1, take C=F itself.
+
+Because U+s_0N is nef with positive-dimensional nef reduction and
+U is semiample, both U and N have degree zero on C: this
+follows from Lemma~lemma:pseudo-vg applied to each, since U is
+nef and N pseudo-effective and their positive combination gives a
+nonnegative, hence zero, degree along F. The strict lift
+widetilde Csubset W avoids the effective exceptional divisors
+E_0,E_1. Both equalities in (eq:two-comparisons) then give
[
p^*Tcdotwidetilde C=p^*Mcdotwidetilde C=0.
]
-The image $p(widetilde C)$ passes through the chosen very general
-point of $X$. Hence $(T+aM)cdot p(widetilde C)>0$ by
+The image p(widetilde C) passes through the chosen very general
+point of X. Hence (T+aM)cdot p(widetilde C)>0 by
(eq:full-condition), but the two equalities above give
-$(T+aM)cdot p(widetilde C)=0$.
+(T+aM)cdot p(widetilde C)=0.
## Positive Iitaka dimension
-Assume (P$_j$) for every $j<n$. Let $(X,B)$ be a projective klt
-$Q$-pair of dimension $n$, let $T=K_X+B$ be pseudo-effective, and
-let $M$ be a nef $Q$-Cartier divisor on $X$. Suppose
-(eq:full-condition) holds for some rational $a>0$, and that
-$kappa(X,T)>0$. Then $T+cM$ is big for every rational $c>0$.
+Assume (P_j) for every j<n. Let (X,B) be a projective klt
+Q-pair of dimension n, let T=K_X+B be pseudo-effective, and
+let M be a nef Q-Cartier divisor on X. Suppose
+(eq:full-condition) holds for some rational a>0, and that
+kappa(X,T)>0. Then T+cM is big for every rational c>0.
The argument reduces to the fibres of the Iitaka fibration of the
@@ -753,57 +739,57 @@ full-nef-dimension divisor of smaller dimension, and the induction
hypothesis applies.
Proof.
-After a small $Q$-factorialization, apply
-Lemma~lemma:two-nef-model to obtain the model $(Y,B_Y)$, its
-transforms $N$, $U$, and the rational $s_0>0$. Let
-$hcolon Yto S$ be the contraction defined by the semiample divisor
-$U$. Since $U$ is semiample, there is an ample rational divisor
-$A_S$ on $S$ with $Usim_Q h^*A_S$, and
-$dim S=kappa(X,T)>0$. Set $Q=U+s_0N$.
-
-A general fibre $F$ of $h$ is a projective variety of dimension
-strictly smaller than $n$ (when positive-dimensional), and $h$ is
-flat over a nonempty open of $S$ after shrinking. The restriction
-pair $(F,B_Y|_F)$ is klt with
+After a small Q-factorialization, apply
+Lemma~lemma:two-nef-model to obtain the model (Y,B_Y), its
+transforms N, U, and the rational s_0>0. Let
+hcolon Yto S be the contraction defined by the semiample divisor
+U. Since U is semiample, there is an ample rational divisor
+A_S on S with Usim_Q h^*A_S, and
+dim S=kappa(X,T)>0. Set Q=U+s_0N.
+
+A general fibre F of h is a projective variety of dimension
+strictly smaller than n (when positive-dimensional), and h is
+flat over a nonempty open of S after shrinking. The restriction
+pair (F,B_Y|_F) is klt with
[
K_F+B_Y|_F sim_Q U|_F sim_Q0.
]
-Moreover, on a very general $F$ the divisor $Q|_F$ is nef of full
-nef dimension: choose a very general point of $F$ outside the
-countable union excluded in the full-nef-dimension condition for $Q$
-on $Y$, and note that such a point exists for every nonempty fibre.
-Since $U|_Fequiv0$, we have $Q|_Fequiv s_0N|_F$, so $N|_F$ is nef
-of full nef dimension. Apply $(P_{dim F})$ to the restriction pair
-$(F,B_Y|_F)$ and the nef divisor $s_0N|_F$; the full condition holds
-for $(N|_F)^{}$ by Lemma~lemma:pseudo-vg. This yields bigness
-of $K_F+B_Y|_F+c(s_0N|_F)$ for every rational $c>0$, and in
-particular of $Q|_F$.
+Moreover, on a very general F the divisor Q|_F is nef of full
+nef dimension: choose a very general point of F outside the
+countable union excluded in the full-nef-dimension condition for Q
+on Y, and note that such a point exists for every nonempty fibre.
+Since U|_Fequiv0, we have Q|_Fequiv s_0N|_F, so N|_F is nef
+of full nef dimension. Apply (P_{dim F}) to the restriction pair
+(F,B_Y|_F) and the nef divisor s_0N|_F; the full condition holds
+for (N|_F)^{} by Lemma~lemma:pseudo-vg. This yields bigness
+of K_F+B_Y|_F+c(s_0N|_F) for every rational c>0, and in
+particular of Q|_F.
We want to pass from a very general fibre to relative bigness over
-$S$. Fix an ample Cartier divisor $H$ on $Y$ and shrink $S$ so that
-$h$ is flat over the chosen open. For every positive integer $ell$
-clearing the denominators of $Q$, the locus
+S. Fix an ample Cartier divisor H on Y and shrink S so that
+h is flat over the chosen open. For every positive integer ell
+clearing the denominators of Q, the locus
[
{sin S: H^0(F_s,OO_{F_s}((ell Q-H)|_{F_s}))neq0}
]
-is closed in $S$, by upper semicontinuity of cohomology in flat
+is closed in S, by upper semicontinuity of cohomology in flat
families. Bigness on a very general fibre puts the base point of
-$h$ into one such locus. If that locus is proper, we have a
+h into one such locus. If that locus is proper, we have a
contradiction with the countability of the possible proper loci; so
-it must be all of $S$. Therefore $ell Q$ is $H+E$ on the generic
-fibre, so $Q$ is relatively big over $S$. By the standard relative
+it must be all of S. Therefore ell Q is H+E on the generic
+fibre, so Q is relatively big over S. By the standard relative
form of Kodaira's lemma (see cite[Lemma 3.2.1]{BCHM}), a
-sufficiently large pullback of the ample $A_S$ turns the relatively
-big $Q$ into a big divisor on $Y$; concretely, since $U$ is the
-pullback of $A_S$,
+sufficiently large pullback of the ample A_S turns the relatively
+big Q into a big divisor on Y; concretely, since U is the
+pullback of A_S,
[
Q+bU=(1+b)U+s_0N
]
-is big for some rational $b>0$.
+is big for some rational b>0.
-By (eq:two-comparisons)(ii), this makes $(1+b)T+s_0M$ big on
-$X$. Finally, fix a rational $c>0$ and choose rational
-$lambda>0$ with $lambda(1+b)<1$ and $lambda s_0<c$. Then
+By (eq:two-comparisons)(ii), this makes (1+b)T+s_0M big on
+X. Finally, fix a rational c>0 and choose rational
+lambda>0 with lambda(1+b)<1 and lambda s_0<c. Then
[
T+cM=
lambda((1+b)T+s_0M)
@@ -817,44 +803,44 @@ big.
## The Calabi--Yau reduction
-Assume (P$_j$) for every $j<n$. Suppose that on every projective
-klt $Q$-pair $(V,Delta)$ of dimension $n$ with $K_V+Deltaequiv0$,
-every nef $Q$-Cartier divisor of full nef dimension is big. Then
-(P$_n$) holds.
+Assume (P_j) for every j<n. Suppose that on every projective
+klt Q-pair (V,Delta) of dimension n with K_V+Deltaequiv0,
+every nef Q-Cartier divisor of full nef dimension is big. Then
+(P_n) holds.
The proposition makes the reduction to
Theorem~thm:key-ingredient explicit: away from the case
-$kappa(T)=0$, the positive-dimensional Iitaka argument of
-Proposition~prop:full-positive already gives $(P_n)$; the only
+kappa(T)=0, the positive-dimensional Iitaka argument of
+Proposition~prop:full-positive already gives (P_n); the only
remaining case is a klt Calabi--Yau pair with a full-nef-dimension
nef summand.
Proof.
-Use a small $Q$-factorialization and
+Use a small Q-factorialization and
Lemma~lemma:two-nef-model. The Iitaka dimension
-$kappa(X,T)ge0$ is either positive, in which case
+kappa(X,T)ge0 is either positive, in which case
Proposition~prop:full-positive applies directly, or zero.
-Suppose $kappa(X,T)=0$. Then $U$ is semiample with Iitaka
-dimension zero, so $Usim_Q0$. The pair $(Y,B_Y)$ is a klt
-Calabi--Yau pair of dimension $n$. The nef divisor $U+s_0N$ has
+Suppose kappa(X,T)=0. Then U is semiample with Iitaka
+dimension zero, so Usim_Q0. The pair (Y,B_Y) is a klt
+Calabi--Yau pair of dimension n. The nef divisor U+s_0N has
full nef dimension by Lemma~lemma:two-nef-model(2), and since
-$Usim_Q0$ we have $U+s_0Nsim_Q s_0N$; thus $N$ is nef of full
-nef dimension. By hypothesis $N$ is big, and so is $s_0N$.
+Usim_Q0 we have U+s_0Nsim_Q s_0N; thus N is nef of full
+nef dimension. By hypothesis N is big, and so is s_0N.
-By (eq:two-comparisons)(ii), the pullback $p^*(T+s_0M)$ is
-the sum of a $q$-exceptional effective divisor and the pullback of
-$U+s_0N$; hence $T+s_0M$ is big on $X$. The positive-combination
+By (eq:two-comparisons)(ii), the pullback p^*(T+s_0M) is
+the sum of a q-exceptional effective divisor and the pullback of
+U+s_0N; hence T+s_0M is big on X. The positive-combination
argument at the end of the proof of Proposition~prop:full-positive
-(which requires only $T$ pseudo-effective and $M$ nef) then gives
-that $T+cM$ is big for every rational $c>0$.
+(which requires only T pseudo-effective and M nef) then gives
+that T+cM is big for every rational c>0.
-What remains for (P$_n$) is therefore the hypothesis of
+What remains for (P_n) is therefore the hypothesis of
Proposition~prop:reduction-cy: on a projective klt Calabi--Yau
-$Q$-pair, a nef $Q$-Cartier divisor of full nef dimension is big.
+Q-pair, a nef Q-Cartier divisor of full nef dimension is big.
This is Theorem~thm:key-ingredient. We prove it first for
-terminal $X$ with $K_Xsim0$ in
+terminal X with K_Xsim0 in
Sections~sec:jet-tools--sec:frobenius, and then extend
to all klt Calabi--Yau pairs in Section~sec:completion.
@@ -870,7 +856,7 @@ pass to a subvariety, so that lower-dimensional induction hypotheses
apply. Third, the LA machinery that produces subvarieties satisfying
those bounds from a jet kernel with positive-dimensional base locus.
-All varieties in this section are over $C$.
+All varieties in this section are over C.
## The bounded-degree obstruction
@@ -880,17 +866,17 @@ arbitrarily thin in degree, when its ambient variety satisfies a
bounded volume and bounded canonical ratio.
[Bounded-degree curves]
-For every integer $fge1$ there is an integer $m_f>0$ with the
-following property. Let $Y$ be a smooth integral projective variety
-of dimension $f$, let $D_0$ be a reduced simple normal crossing
-divisor or the zero divisor, and let $L_0$ be a nef integral Cartier
-divisor of full nef dimension on $Y$. Suppose that $P_0$ is a nef
-rational Cartier divisor, and that $r,C_0,C_1$ are positive real
+For every integer fge1 there is an integer m_f>0 with the
+following property. Let Y be a smooth integral projective variety
+of dimension f, let D_0 be a reduced simple normal crossing
+divisor or the zero divisor, and let L_0 be a nef integral Cartier
+divisor of full nef dimension on Y. Suppose that P_0 is a nef
+rational Cartier divisor, and that r,C_0,C_1 are positive real
numbers with
[
- 0<P_0^fle C_0,qquad P_0-rL_0 is nef.
+ 0<P_0^fle C_0, P_0-rL_0 is nef.
]
-Suppose also $A_0=K_Y+D_0+L_0$ is big and
+Suppose also A_0=K_Y+D_0+L_0 is big and
A_0cdot P_0^{f-1}le C_1 P_0^f.
@@ -898,75 +884,75 @@ Then
rle(m_f C_1)^{f-1} C_0.
-The integer $m_f$ depends only on $f$ and not on the variety or the
-divisors. For all $1le fle n$, one may take a single integer $m$
-depending only on $n$.
+The integer m_f depends only on f and not on the variety or the
+divisors. For all 1le fle n, one may take a single integer m
+depending only on n.
-The quantity $m_f$ is not explicit: it comes from the Birkar--Zhang
+The quantity m_f is not explicit: it comes from the Birkar--Zhang
effective birationality theorem for polarized pairs
cite[Theorem~1.3]{BZ}. What matters is that it depends only on the
dimension, not on the variety or the divisors.
Proof.
-The pair $(Y,D_0)$ is log canonical with boundary coefficients in
-${0,1}$, and the polarizing divisor $L_0$ is a nef integral Cartier
-divisor. By cite[Theorem~1.3]{BZ}, an integer $m_f$ exists such
-that $|m_f A_0|$ is birational. Because $A_0$ is integral Cartier,
+The pair (Y,D_0) is log canonical with boundary coefficients in
+{0,1}, and the polarizing divisor L_0 is a nef integral Cartier
+divisor. By cite[Theorem~1.3]{BZ}, an integer m_f exists such
+that |m_f A_0| is birational. Because A_0 is integral Cartier,
no rounding or additional denominator is involved.
Resolve the base ideal by a projective birational morphism
-$mucolonwidetilde Yto Y$ with $widetilde Y$ smooth, and write
+mucolonwidetilde Yto Y with widetilde Y smooth, and write
[
mu^*(m_f A_0)=H+F
]
-for the basepoint-free moving part $H$ and an effective $F$. The
-map defined by $H$ is birational onto its image; $H$ is nef and big.
-In the rest of this proof, $P_0$ and $L_0$ denote pullbacks to
-$widetilde Y$. Effectivity of $F$ and nefness of $P_0$ give
+for the basepoint-free moving part H and an effective F. The
+map defined by H is birational onto its image; H is nef and big.
+In the rest of this proof, P_0 and L_0 denote pullbacks to
+widetilde Y. Effectivity of F and nefness of P_0 give
Hcdot P_0^{f-1}le m_f C_1 P_0^f.
-For $fge2$, put $s_i=P_0^{f-i}cdot H^i$ for $0le ile f$.
-Both divisors are nef and big, so each $s_i$ is positive. The
+For fge2, put s_i=P_0^{f-i}cdot H^i for 0le ile f.
+Both divisors are nef and big, so each s_i is positive. The
Khovanskii--Teissier inequalities
cite[Corollary~1.6.3(i) and Example~1.6.4]{LazarsfeldPAGI}, namely
-$s_i^2ge s_{i-1}s_{i+1}$, show that the ratios $s_i/s_{i-1}$ are
+s_i^2ge s_{i-1}s_{i+1}, show that the ratios s_i/s_{i-1} are
decreasing; hence
P_0cdot H^{f-1}
le P_0^f(frac{Hcdot P_0^{f-1}}{P_0^f})^{f-1}
le (m_f C_1)^{f-1} C_0.
-Choose a very general point $yinwidetilde Y$ in the isomorphism
-open of $mu$, and such that $mu(y)$ is very general with respect
-to the full-nef-dimension condition on $L_0$. Pull back $f-1$
-general hyperplanes through the image of $y$ under the morphism
-defined by $H$. This subsystem has only the base point $y$, so its
+Choose a very general point yinwidetilde Y in the isomorphism
+open of mu, and such that mu(y) is very general with respect
+to the full-nef-dimension condition on L_0. Pull back f-1
+general hyperplanes through the image of y under the morphism
+defined by H. This subsystem has only the base point y, so its
general members meet in an effective curve cycle with class
-$H^{f-1}$; the component $C$ through $y$ is a curve because the
-local equations are independent. Nefness of $P_0$ gives
+H^{f-1}; the component C through y is a curve because the
+local equations are independent. Nefness of P_0 gives
[
P_0cdot Cle P_0cdot H^{f-1}.
]
-The curve $C$ is not contracted by $mu$, and its image passes
-through the very general point $mu(y)$. Full nef dimension and
-integrality therefore give $L_0cdot Cge1$. Since $P_0-rL_0$ is
-nef, we obtain $P_0cdot Cge r$, which together with
+The curve C is not contracted by mu, and its image passes
+through the very general point mu(y). Full nef dimension and
+integrality therefore give L_0cdot Cge1. Since P_0-rL_0 is
+nef, we obtain P_0cdot Cge r, which together with
(j:mixed-degree) proves (j:bounded-curve-obstruction).
-If $f=1$, take $C=Y$ directly; then
-$P_0cdot Yge rL_0cdot Yge r$, and $rle C_0$ is the claim.
+If f=1, take C=Y directly; then
+P_0cdot Yge rL_0cdot Yge r, and rle C_0 is the claim.
Finally, cite[Theorem~1.3]{BZ} applies to every multiple of its
prescribed integer; a common multiple of the finitely many integers
-$m_1,...,m_n$ gives the last assertion.
+m_1,...,m_n gives the last assertion.
-The lemma is uniform in the sense that varieties, divisors, and $r$
-may vary, provided $C_0$ and $C_1$ stay bounded. In the geometric
-section, $rto∞$ along a family, so a sequence satisfying the
-hypotheses of the lemma will automatically be absent for large $r$.
+The lemma is uniform in the sense that varieties, divisors, and r
+may vary, provided C_0 and C_1 stay bounded. In the geometric
+section, rto∞ along a family, so a sequence satisfying the
+hypotheses of the lemma will automatically be absent for large r.
## Very general images
@@ -975,35 +961,35 @@ and pseudo-effectivity of the canonical divisor are compatible with
passing to sufficiently general subvarieties.
[Very general images]
-Let $X$ be a normal integral projective variety that is not
-uniruled, and let $L$ be a nef Cartier divisor of full nef dimension
-on $X$. Then there is a complement $X^circ$ of a countable union of
-proper closed subsets of $X$ with the following property. If $Y$ is
-a smooth integral projective variety and $gcolon Yto X$ is
+Let X be a normal integral projective variety that is not
+uniruled, and let L be a nef Cartier divisor of full nef dimension
+on X. Then there is a complement X^circ of a countable union of
+proper closed subsets of X with the following property. If Y is
+a smooth integral projective variety and gcolon Yto X is
generically finite onto a positive-dimensional image that meets
-$X^circ$, then $g^*L$ has full nef dimension, $Y$ is not uniruled,
-and $K_Y$ is pseudo-effective.
+X^circ, then g^*L has full nef dimension, Y is not uniruled,
+and K_Y is pseudo-effective.
The point is that very general conditions simultaneously test full
-nef dimension of $L$ and avoid all rational curves.
+nef dimension of L and avoid all rational curves.
Proof.
-Choose $X^circ$ so that every curve through any of its points has
-positive $L$-degree and no rational curve meets $X^circ$. The first
+Choose X^circ so that every curve through any of its points has
+positive L-degree and no rational curve meets X^circ. The first
condition characterizes full nef dimension at a point. For the
second, rational curves move in countably many algebraic families;
the closure of the evaluation image of each such family is proper
-because $X$ is not uniruled. Removing those closures gives $X^circ$.
+because X is not uniruled. Removing those closures gives X^circ.
-Since $g(Y)$ meets $X^circ$, it is contained in none of the closed
-sets excluded by the definition. A very general point of $Y$ maps
-into $X^circ$ and lies in the quasi-finite locus of $g$. Every
+Since g(Y) meets X^circ, it is contained in none of the closed
+sets excluded by the definition. A very general point of Y maps
+into X^circ and lies in the quasi-finite locus of g. Every
curve through such a point has a curve as its image, so its
-$g^*L$-degree is positive. This gives full nef dimension of $g^*L$.
-A covering family of rational curves on $Y$ would similarly give a
-nonconstant rational curve meeting $X^circ$, a contradiction; so
-$Y$ is not uniruled. Pseudo-effectivity of $K_Y$ now follows from
+g^*L-degree is positive. This gives full nef dimension of g^*L.
+A covering family of rational curves on Y would similarly give a
+nonconstant rational curve meeting X^circ, a contradiction; so
+Y is not uniruled. Pseudo-effectivity of K_Y now follows from
[cite].
@@ -1021,26 +1007,26 @@ subvariety are computed after pullback to its normalization and a
smooth projective resolution, because the subvariety itself may be
singular.
-A linear system $mathcal HcalHsubset H^0(Z,OO_Z(kR))$ has a
-base ideal $mathfrak b$frakb$ defined as the ideal generated by its
-sections in $OO_Z$; its zero locus is the base locus. A base
-component $V$ is isolated at the generic point of $V$ when the
-localized ideal $mathfrak bfrakbOO_{Z,eta_V}$ is primary for the
-maximal ideal $mathfrak m$frakm$ of $OO_{Z,eta_V}$. In this case there
-is a well-defined multiplicity $h$ with
-$mathfrak bfrakbOO_{Z,eta_V}subsetmathfrak m^h$,frakm^h$, measuring how
-strongly $mathcal H$calH$ vanishes along $V$ at its generic point.
+A linear system mathcal HcalHsubset H^0(Z,OO_Z(kR)) has a
+base ideal mathfrak bfrakb defined as the ideal generated by its
+sections in OO_Z; its zero locus is the base locus. A base
+component V is isolated at the generic point of V when the
+localized ideal mathfrak bfrakbOO_{Z,eta_V} is primary for the
+maximal ideal mathfrak mfrakm of OO_{Z,eta_V}. In this case there
+is a well-defined multiplicity h with
+mathfrak bfrakbOO_{Z,eta_V}subsetmathfrak m^h,frakm^h, measuring how
+strongly mathcal HcalH vanishes along V at its generic point.
The first input is a numerical version of the
Kawakita--Ambro adjunction for a lc-trivial fibration:
[Numerical generic subadjunction, {cite[Lemma~7.1]{LA}}]
-Let $Z$ be a projective $Q$-factorial klt variety, let $Θge0$
-be a rational divisor, and let $Vsubset Z$ be integral of dimension
-$f>0$. Suppose $(Z,Θ)$ is log canonical at the generic point of
-$V$, and admits a divisor of log discrepancy zero centered on $V$. If
-$P$ is a nef rational Cartier divisor on $V$ and $rhocolon V^*to V$
+Let Z be a projective Q-factorial klt variety, let Θge0
+be a rational divisor, and let Vsubset Z be integral of dimension
+f>0. Suppose (Z,Θ) is log canonical at the generic point of
+V, and admits a divisor of log discrepancy zero centered on V. If
+P is a nef rational Cartier divisor on V and rhocolon V^*to V
is a smooth projective resolution factoring through its
normalization, then
[
@@ -1050,51 +1036,51 @@ normalization, then
Only generic log canonicity is required: the inequality lives at the
-generic point of $V$. The proof in [cite] passes through the dlt
+generic point of V. The proof in [cite] passes through the dlt
modification of cite[Theorem~2.10]{FH}, whose truncated boundary
leaves an effective surplus supported over the non-lc locus. On a
-minimal stratum over $V$, adjunction gives a pair that is klt over
+minimal stratum over V, adjunction gives a pair that is klt over
the generic point of its connected-fiber base. This suffices for the
klt-trivial canonical bundle formula and its b-nef moduli divisor
cite[Definition~3.1 and Theorem~3.5]{FG}. The effective
discriminant, the moduli divisor, and finite ramification all have
nonnegative intersections with the pulled-back nef class. The
comparison uses canonical Weil cycles, so it requires no
-$Q$-Gorenstein assumption on the normalization of $V$.
+Q-Gorenstein assumption on the normalization of V.
The second input transfers a base component to a genuine pair:
[High-multiplicity base component, {cite[Lemma~7.2]{LA}}]
-Let $Z$ be a projective $Q$-factorial klt variety of dimension $d$,
-let $R$ be a nef rational Cartier divisor, and let $k>0$ be an integer
-such that $kR$ is Cartier. Let
-$0nemathcal HcalHsubset H^0(Z,OO_Z(kR))$ be a linear subspace with
-proper base locus and base ideal $mathfrak b$.frakb$. Suppose that
-$V$ is an integral base-locus component of dimension $f<d$, isolated
-at its generic point $eta_V$, and that $eta_Vin Z_{sm}$.
-Let $mathfrak m$frakm$ be the maximal ideal of $OO_{Z,eta_V}$. If
-$mathfrak bfrakbOO_{Z,eta_V}subsetmathfrak m^h$frakm^h$ for an integer $h>0$,
-there are a rational $0<cle(d-f)k/h$ and an effective rational
-divisor $Θsim_Q cR$ such that $(Z,Θ)$ is log canonical
-at $eta_V$ and has a log canonical place centered on $V$. The
+Let Z be a projective Q-factorial klt variety of dimension d,
+let R be a nef rational Cartier divisor, and let k>0 be an integer
+such that kR is Cartier. Let
+0nemathcal HcalHsubset H^0(Z,OO_Z(kR)) be a linear subspace with
+proper base locus and base ideal mathfrak b.frakb. Suppose that
+V is an integral base-locus component of dimension f<d, isolated
+at its generic point eta_V, and that eta_Vin Z_{sm}.
+Let mathfrak mfrakm be the maximal ideal of OO_{Z,eta_V}. If
+mathfrak bfrakbOO_{Z,eta_V}subsetmathfrak m^hfrakm^h for an integer h>0,
+there are a rational 0<cle(d-f)k/h and an effective rational
+divisor Θsim_Q cR such that (Z,Θ) is log canonical
+at eta_V and has a log canonical place centered on V. The
degree and canonical bounds are
R^fcdot V&le(k/h)^{d-f}R^d,
K_{V^*}cdot R^{f-1}
- &le(K_Z+cR)|_Vcdot R^{f-1}qquad(f>0),
+ &le(K_Z+cR)|_Vcdot R^{f-1}(f>0),
-where $V^*$ is a smooth projective resolution. The degree bound
-also holds at $f=0$.
+where V^* is a smooth projective resolution. The degree bound
+also holds at f=0.
-The degree estimate is the local multiplicity bound $h^{d-f}$
+The degree estimate is the local multiplicity bound h^{d-f}
combined with successive general cuts by the given linear series.
The boundary is obtained by averaging general members with total
coefficient equal to the generic log canonical threshold of
-$mathfrak b$;frakb$; thus it is an actual divisor to which
+mathfrak b;frakb; thus it is an actual divisor to which
Lemma~j:subadjunction applies.
The third input produces the moving base components in the first
@@ -1102,57 +1088,57 @@ place:
[Moving multiplicity, {cite[Lemma~7.3]{LA}}]
-Let $Z$ be an integral projective variety of dimension $d$, let $R$
+Let Z be an integral projective variety of dimension d, let R
be a nef, big, semiample rational Cartier divisor, and fix an integer
-$k>0$ such that $kR$ is Cartier. Choose positive real numbers
-$tau_0<...<tau_d$ with common gap $delta>0$. For $xin
-Z_{sm}$, put
+k>0 such that kR is Cartier. Choose positive real numbers
+tau_0<...<tau_d with common gap delta>0. For xin
+Z_{sm}, put
[
mathcal H_i(x)=H^0(Z,calH_i(x)=H^0(Z,OO_Z(kR)
⊗ mathfrak m_x^frakm_x^{lceil ktau_irceil}).
]
-Suppose that for very general $x$ all these spaces are nonzero, and
-that the base locus of $mathcal H_0(x)$calH_0(x)$ has a positive-dimensional
-component through $x$. If $kdeltage4$, then, after an algebraic
+Suppose that for very general x all these spaces are nonzero, and
+that the base locus of mathcal H_0(x)calH_0(x) has a positive-dimensional
+component through x. If kdeltage4, then, after an algebraic
parameter extension and restriction to nonempty open subsets, there
-are: an integral parameter variety $T$ with a dominant marked-point
-map $xcolon Tto Z_{sm}$; an index $iin{0,...,d-1}$
-fixed throughout; and a family of integral subvarieties $V_t$
-containing $x(t)$ and common to the base loci of $mathcal
-H_i(x(t))$calH_i(x(t))$ and $mathcal H_calH_{i+1}(x(t))$. The incidence
+are: an integral parameter variety T with a dominant marked-point
+map xcolon Tto Z_{sm}; an index iin{0,...,d-1}
+fixed throughout; and a family of integral subvarieties V_t
+containing x(t) and common to the base loci of mathcal
+H_i(x(t))calH_i(x(t)) and mathcal H_calH_{i+1}(x(t)). The incidence
[
Gamma={(t,z):zin V_t}subset Ttimes Z
]
-dominates $Z$.
+dominates Z.
-For general $t$, the dimension $f$ of $V_t$ lies in $[1,d-1]$,
-$V_t$ is an isolated component of the higher base locus at its
+For general t, the dimension f of V_t lies in [1,d-1],
+V_t is an isolated component of the higher base locus at its
generic point, and the entire higher-kernel base ideal is contained
-there in the ordinary power $mathcal I_calI_{V_t}^h$, where
+there in the ordinary power mathcal I_calI_{V_t}^h, where
[
- h=lceil kdelta/2rceil,qquad k/hle2/delta.
+ h=lceil kdelta/2rceil, k/hle2/delta.
]
The incidence base ideal is likewise contained in
-$mathcal I_calI_Gamma^h$ on a dense open of $Gamma$. These
-containments are taken in the smooth ambient open. If $Z$ is also
-$Q$-factorial and klt, then
+mathcal I_calI_Gamma^h on a dense open of Gamma. These
+containments are taken in the smooth ambient open. If Z is also
+Q-factorial and klt, then
R^fcdot V_t&le(2/delta)^{d-f}R^d,
K_{V_t^*}cdot R^{f-1}
- &le(K_Z+cR)|_{V_t}cdot R^{f-1},qquad
+ &le(K_Z+cR)|_{V_t}cdot R^{f-1},
0<cle2(d-f)/delta.
-The top intersection on $V_t$ is positive when $V_t$ meets the
+The top intersection on V_t is positive when V_t meets the
open on which the big semiample contraction is an isomorphism.
The use of full kernels matters. Differentiation in the parameter
lowers vanishing at the moving mark and still gives a global section
in the lower kernel. Incidence dominance makes the parameter
-directions span the normal directions to $V_t$, and the vanishing of
-all derivatives of order less than $h$ gives the ordinary-power
+directions span the normal directions to V_t, and the vanishing of
+all derivatives of order less than h gives the ordinary-power
containment.
The fourth input restricts an isolated base ideal to a fibre of a
@@ -1161,27 +1147,27 @@ morphism:
[Restriction of an isolated base ideal,
{cite[Lemma~7.4]{LA}}]
-In the setting of Lemma~j:moving-kernel, let $gcolon Zto B$
+In the setting of Lemma~j:moving-kernel, let gcolon Zto B
be a morphism. There is a dense open of the incidence with the
-following property. Choose $(t,z)$ in this open, put $b=g(z)$, and
-suppose $V_t$ is smooth over a smooth open of its actual image at
-$z$. Let $F$ be the reduced component through $z$ of the scheme
-fibre of $V_tto g(V_t)$. Near $z$ in $Z_b$, the restricted
-higher-kernel base ideal has support exactly $F$ and is contained
-in the ordinary power $mathcal I_F^h$.calI_F^h$.
-
-If $F$ is proper in an integral ambient fibre component containing
-it, the restricted series is nonzero there and $F$ is an isolated
+following property. Choose (t,z) in this open, put b=g(z), and
+suppose V_t is smooth over a smooth open of its actual image at
+z. Let F be the reduced component through z of the scheme
+fibre of V_tto g(V_t). Near z in Z_b, the restricted
+higher-kernel base ideal has support exactly F and is contained
+in the ordinary power mathcal I_F^h.calI_F^h.
+
+If F is proper in an integral ambient fibre component containing
+it, the restricted series is nonzero there and F is an isolated
base component at its generic point. Applying
Lemma~j:base-component on that ambient component additionally
-requires its projectivity, $Q$-factoriality, klt singularities, and
-smoothness at the generic point of $F$.
+requires its projectivity, Q-factoriality, klt singularities, and
+smoothness at the generic point of F.
The fibre is chosen through a general incidence point; the statement
does not assert the same conclusion for an arbitrary special fibre.
Smoothness over the actual image identifies the restricted ideal of
-$V_t$ with the reduced ideal of $F$ locally, preserving ordinary
+V_t with the reduced ideal of F locally, preserving ordinary
powers.
The fifth input turns a bounded-degree family of finite covers into
@@ -1190,50 +1176,50 @@ a finite list once the branch divisors do not sweep:
[Finite covers after excluding sweeping branch divisors,
{cite[Lemma~7.5]{LA}}]
-Let $Y$ be a normal integral projective variety, let
-$mathcal VcalVto T$ be a smooth projective family with integral fibres
-over an integral parameter variety, and let $gcolonmathcal VcalVto Y$
+Let Y be a normal integral projective variety, let
+mathcal VcalVto T be a smooth projective family with integral fibres
+over an integral parameter variety, and let gcolonmathcal VcalVto Y
be a morphism. Suppose the general fibre maps
-$g_tcolon V_tto Y$ are dominant and generically finite of degree
-at most a fixed integer $e$.
+g_tcolon V_tto Y are dominant and generically finite of degree
+at most a fixed integer e.
After a finite parameter extension and shrinking, spread the
geometric generic components of the zero divisor of the relative
-Jacobian over $Y_{sm}$ to divisors
-$mathcal R_1,...,calR_1,...,mathcal R_N$calR_N$ in $mathcal V$.calV$. Retain this
-notation after the base change. Let $I$ consist of those $j$ for
-which $g_t((mathcal R_j)_t)bar$calR_j)_t)bar$ is a divisor in $Y$ for
-general $t$. Assume
+Jacobian over Y_{sm} to divisors
+mathcal R_1,...,calR_1,...,mathcal R_NcalR_N in mathcal V.calV. Retain this
+notation after the base change. Let I consist of those j for
+which g_t((mathcal R_j)_t)barcalR_j)_t)bar is a divisor in Y for
+general t. Assume
[
g(mathcal R_j)barcalR_j)barne Yqquad(jin I).
]
After shrinking again, there is a smooth nonempty open
-$Y^circsubset Y$ over which every general finite normal Stein
-cover of $g_t$ is finite 'etale. Only finitely many such normal
-covers of $Y$ occur, up to isomorphism over $Y$. The conclusion
+Y^circsubset Y over which every general finite normal Stein
+cover of g_t is finite 'etale. Only finitely many such normal
+covers of Y occur, up to isomorphism over Y. The conclusion
holds simultaneously for finitely many projections; the open and the
finite lists may depend on all fixed parameters of the family.
-The hypothesis removes $Y_{sing}$ and the finitely many
-proper image closures $g(mathcal R_j)bar$calR_j)bar$ for $jin I$.
+The hypothesis removes Y_{sing} and the finitely many
+proper image closures g(mathcal R_j)barcalR_j)bar for jin I.
Every branch prime of a general Stein cover is accounted for by one
of these relative Jacobian components, so purity makes the
restricted covers 'etale. The fundamental group of the resulting
smooth complex variety is finitely generated, so it has only
-finitely many covers of degree at most $e$. Finally, a finite
-normal cover of $Y$ is determined by its function field, hence by
+finitely many covers of degree at most e. Finally, a finite
+normal cover of Y is determined by its function field, hence by
its restriction to this fixed open.
-## $Q$-factorialization of a klt ambient variety
+## Q-factorialization of a klt ambient variety
-The base-component estimates require a $Q$-factorial ambient
+The base-component estimates require a Q-factorial ambient
variety. When the ambient is only klt, we may pass to a small
-projective $Q$-factorialization $sigmacolon Z'to Z$. This
+projective Q-factorialization sigmacolon Z'to Z. This
passage is made only for components that meet the isomorphism open of
-$sigma$. The strict transforms then retain the generic
+sigma. The strict transforms then retain the generic
base-ideal and multiplicity conditions for the pulled-back series,
-and $sigma^*R$ is nef. Since $K_{Z'}=sigma^*K_Z$, the projection
+and sigma^*R is nef. Since K_{Z'}=sigma^*K_Z, the projection
formula gives the same degree and canonical-intersection bounds on a
common resolution. The pulled-back testing class need not be
ample; nefness is the property used in these estimates. Passing to
@@ -1250,70 +1236,70 @@ with trivial canonical divisor. This is where the geometry of the
paper is concentrated.
[Terminal Calabi--Yau case]
-Let $nge2$ and assume $(P_j)$ for every $j<n$. Let $X$ be a
-projective $Q$-factorial terminal complex variety of dimension $n$
-with $K_Xsim0$. If $L$ is a nef Cartier divisor on $X$ of full
-nef dimension, then $L$ is big.
+Let nge2 and assume (P_j) for every j<n. Let X be a
+projective Q-factorial terminal complex variety of dimension n
+with K_Xsim0. If L is a nef Cartier divisor on X of full
+nef dimension, then L is big.
The proof of Theorem~thm:terminal-big occupies this section and
-Section~sec:frobenius. Assume $L$ is not big. We produce two
+Section~sec:frobenius. Assume L is not big. We produce two
incompatible jet estimates: a vanishing-order bound for sections of
big multiples of a rescaled ample approximation (Lemma~lemma:scalar-bound),
-and a large jet system on a projective bundle over $Xtimes X$
+and a large jet system on a projective bundle over Xtimes X
(Proposition~prop:large-seshadri). The incompatibility is made
precise by the Frobenius comparison of Section~sec:frobenius.
## Normalization and the curve obstruction
-Throughout, fix $X,L$ forming a counterexample to
-Theorem~thm:terminal-big. Thus: $X$ is projective,
-$Q$-factorial, terminal, of dimension $nge2$, with $K_Xsim0$; and
-$L$ is a nef Cartier divisor of full nef dimension but not big. In
-particular, $L^n=0$, and every curve through a very general point of
-$X$ has positive integral $L$-degree.
+Throughout, fix X,L forming a counterexample to
+Theorem~thm:terminal-big. Thus: X is projective,
+Q-factorial, terminal, of dimension nge2, with K_Xsim0; and
+L is a nef Cartier divisor of full nef dimension but not big. In
+particular, L^n=0, and every curve through a very general point of
+X has positive integral L-degree.
-Fix an ample Cartier divisor $A$ on $X$ and choose a rational
-$ε>0$ with
+Fix an ample Cartier divisor A on X and choose a rational
+ε>0 with
- (14ε)^n<tfrac14, qquad 80ε<tfrac34.
+ (14ε)^n<tfrac14, 80ε<tfrac34.
-Let $t>0$ tend to zero through positive rationals. Since $L$ is nef
-and not big, $L^n=0$; we choose positive integers $r=r(t)$ by
+Let t>0 tend to zero through positive rationals. Since L is nef
+and not big, L^n=0; we choose positive integers r=r(t) by
- P=P_t=r(L+tA),qquad
- r→∞,qquad P^n→ε^n.
+ P=P_t=r(L+tA),
+ r→∞, P^n→ε^n.
-For instance, $r$ may be the nearest integer to
-$ε/((L+tA)^n)^{1/n}$; if the expansion of $(L+tA)^n$ is
-nonzero at every order up to $t^{n-1}$ (and it is, since
-$L^n=0$ and $L^{n-1}A>0$), this makes $rto∞$ as $tto0$.
-The divisor $P$ is ample and rational Cartier, and $P-rL=rtA$ is nef
-for all $t$. After discarding finitely many $t$ we may assume
+For instance, r may be the nearest integer to
+ε/((L+tA)^n)^{1/n}; if the expansion of (L+tA)^n is
+nonzero at every order up to t^{n-1} (and it is, since
+L^n=0 and L^{n-1}A>0), this makes rto∞ as tto0.
+The divisor P is ample and rational Cartier, and P-rL=rtA is nef
+for all t. After discarding finitely many t we may assume
ε^n/2le P^nle 2ε^n.
-Every curve through a very general point of $X$ has positive
-integral $L$-degree, hence $P$-degree at least $r$ (on such a curve,
-$Pcdot C = r(L+tA)cdot C ge r$ because $Lcdot Cge1$ and
-$Acdot Cge0$). Therefore a covering family of curves of bounded
-$P$-degree will contradict full nef dimension.
+Every curve through a very general point of X has positive
+integral L-degree, hence P-degree at least r (on such a curve,
+Pcdot C = r(L+tA)cdot C ge r because Lcdot Cge1 and
+Acdot Cge0). Therefore a covering family of curves of bounded
+P-degree will contradict full nef dimension.
We will use the same obstruction on subvarieties and covers, so we
record how it descends.
-A smooth projective resolution $Wto X$ admits an effective
+A smooth projective resolution Wto X admits an effective
canonical divisor: pull back a nowhere-zero canonical form from the
-smooth locus of $X$, and use terminality of $X$ to represent
-$K_W$ by an effective $L$-exceptional divisor. In particular $W$,
-and therefore $X$, is not uniruled [cite].
+smooth locus of X, and use terminality of X to represent
+K_W by an effective L-exceptional divisor. In particular W,
+and therefore X, is not uniruled [cite].
-By Lemma~j:very-general-images, if $Y$ is a smooth projective
-variety with a generically finite map $gcolon Yto X$ onto a
+By Lemma~j:very-general-images, if Y is a smooth projective
+variety with a generically finite map gcolon Yto X onto a
positive-dimensional image that meets a sufficiently very general
-point of $X$, then $Y$ is not uniruled, $K_Y$ is pseudo-effective,
-and the pullback $L_Y=g^*L$ has full nef dimension. If
-$f=dim Y<n$, the induction hypothesis $(P_f)$ applied to $(Y,0)$
+point of X, then Y is not uniruled, K_Y is pseudo-effective,
+and the pullback L_Y=g^*L has full nef dimension. If
+f=dim Y<n, the induction hypothesis (P_f) applied to (Y,0)
gives
K_Y+L_Y is big.
@@ -1321,185 +1307,185 @@ gives
All uses of ``very general'' below include the fixed conditions of
Lemma~j:very-general-images.
-Consequently, for $f<n$ it suffices to find a nef testing divisor
-$P_Y$ on such $Y$ with
+Consequently, for f<n it suffices to find a nef testing divisor
+P_Y on such Y with
[
- 0<P_Y^fle C_0,qquad P_Y-rL_Y nef,qquadand
+ 0<P_Y^fle C_0, P_Y-rL_Y nef,qquadand
K_YP_Y^{f-1}le C, P_Y^f,
]
-where $C_0$ and $C$ are independent of $t$. Indeed
-$L_YP_Y^{f-1}le P_Y^f/r$, so Lemma~lemma:bounded-curves
-applies. We use that lemma in dimension $n$ as well, after proving
+where C_0 and C are independent of t. Indeed
+L_YP_Y^{f-1}le P_Y^f/r, so Lemma~lemma:bounded-curves
+applies. We use that lemma in dimension n as well, after proving
the needed bigness separately.
-Call a bound uniform if it does not depend on $t$; it may
-depend on parameters fixed before $t$ tends to zero. Since
-dimensions lie in the finite range $[1,n]$, Lemma~lemma:bounded-curves
-excludes all such configurations once $r$ is large enough.
+Call a bound uniform if it does not depend on t; it may
+depend on parameters fixed before t tends to zero. Since
+dimensions lie in the finite range [1,n], Lemma~lemma:bounded-curves
+excludes all such configurations once r is large enough.
## The scalar jet estimate
-Put $rho=(P^n)^{1/n}$. The first incompatible estimate is the
+Put rho=(P^n)^{1/n}. The first incompatible estimate is the
following vanishing-order bound. It is stronger than a bound on a
single linear system.
In the setting of (eq:scaling), for all sufficiently small
-$t$, at a very general point $xin X$, every nonzero section $s$ of
-every $kP$ with $k$ a positive integer and $kP$ Cartier satisfies
+t, at a very general point xin X, every nonzero section s of
+every kP with k a positive integer and kP Cartier satisfies
ord_x(s)le 4krho.
Proof.
-Suppose violations occur for arbitrarily small $t$. Fix such a $t$.
-At a very general point, a violation for some $k$ means the
+Suppose violations occur for arbitrarily small t. Fix such a t.
+At a very general point, a violation for some k means the
corresponding complete jet kernel is nonzero at the general point.
Indeed the kernel ranks are algebraic rank conditions on the smooth
-locus of $X$, and there are only countably many choices of $k$ and
+locus of X, and there are only countably many choices of k and
of the required integral order. Raising a violating section to
-powers makes $k$ arbitrarily large and divisible.
+powers makes k arbitrarily large and divisible.
-Choose $n+1$ levels
+Choose n+1 levels
[
tau_i=frac32rho+ideltaquad(0le ile n),
qquaddelta=(2rho)/(n).
]
-These lie strictly between $rho$ and $4rho$. For a sufficiently
-large divisible $k$, the corresponding full jet kernels are nonzero
-and $kdeltage4$.
+These lie strictly between rho and 4rho. For a sufficiently
+large divisible k, the corresponding full jet kernels are nonzero
+and kdeltage4.
Consider the lowest kernel. If its base locus at the marked point
were zero-dimensional, the isolated base component at that point
would have multiplicity at least
-$h_0=lceil ktau_0rceil$. The zero-dimensional case of
-Lemma~j:base-component (its degree bound, valid at $f=0$) would
+h_0=lceil ktau_0rceil. The zero-dimensional case of
+Lemma~j:base-component (its degree bound, valid at f=0) would
then force
[
1le(k/h_0)^nP^n < 1,
]
contradiction: the strict inequality follows because
-$k/h_0le 1/tau_0 = 2/(3rho)$, so
-$(k/h_0)^nP^nle(2/3)^n(P^n/rho^n)=(2/3)^n<1$, since $nge2$.
+k/h_0le 1/tau_0 = 2/(3rho), so
+(k/h_0)^nP^nle(2/3)^n(P^n/rho^n)=(2/3)^n<1, since nge2.
Hence the lowest base locus has a positive-dimensional component
through its marked point.
Lemma~j:moving-kernel now supplies a moving family of proper
-base components $V$ with dominant incidence and dimension
-$1le f<n$. On a resolution $V^*$, let $P_V,L_V$ be the
+base components V with dominant incidence and dimension
+1le f<n. On a resolution V^*, let P_V,L_V be the
restrictions followed by pullback. The lemma gives
[
0<P_V^fle(2/delta)^{n-f}P^n,
- qquad
+
K_{V^*}P_V^{f-1}le cP_V^f,
- qquad
+
0<cle(2(n-f))/(delta).
]
Here the canonical term of the ambient vanishes because
-$K_Xsim0$. The volume window (j:volume-window) bounds
-$delta$ above and away from zero, so these are uniform estimates.
-The incidence point may be chosen over a very general point of $X$;
-then Lemma~j:very-general-images makes $V^*$ non-uniruled
-and $K_{V^*}$ pseudo-effective, with $L_V$ of full nef dimension.
-Equation~(j:lower-big) gives bigness of $K_{V^*}+L_V$, and
-$P_V-rL_V$ is nef; Lemma~lemma:bounded-curves contradicts
-$rto∞$.
+K_Xsim0. The volume window (j:volume-window) bounds
+delta above and away from zero, so these are uniform estimates.
+The incidence point may be chosen over a very general point of X;
+then Lemma~j:very-general-images makes V^* non-uniruled
+and K_{V^*} pseudo-effective, with L_V of full nef dimension.
+Equation~(j:lower-big) gives bigness of K_{V^*}+L_V, and
+P_V-rL_V is nef; Lemma~lemma:bounded-curves contradicts
+rto∞.
## The projective bundle and its volume
We now seek a jet system whose order is much larger than the
-scalar bound $4rho$. The extra positivity comes from the fibres of
-a projective bundle over $Xtimes X$ that carries the two copies of
-$X$ in its two summands. For a divisor $D$ on $X$, we write
-$D_1,D_2$ for its pullback from the two factors of $Xtimes X$.
+scalar bound 4rho. The extra positivity comes from the fibres of
+a projective bundle over Xtimes X that carries the two copies of
+X in its two summands. For a divisor D on X, we write
+D_1,D_2 for its pullback from the two factors of Xtimes X.
Set
Z=PP(OO_X(L)_1 ⊕ OO_X(L)_2)
xrightarrow{pi}Xtimes X,
qquadxi=c_1(OO_Z(1)),
- qquad R=P_1+P_2+qxi,
+ R=P_1+P_2+qxi,
-where $q$ is a positive integer to be fixed shortly. We use the
+where q is a positive integer to be fixed shortly. We use the
convention
[
pi_*OO_Z(a)=Sym^a(OO_X(L)_1 ⊕ OO_X(L)_2)
- quad(age0).
+ (age0).
]
-The tautological class $xi$ is nef, and $R$ is ample. As a
-projective bundle over $Xtimes X$ with klt base, and with an
+The tautological class xi is nef, and R is ample. As a
+projective bundle over Xtimes X with klt base, and with an
unramified flag or torus-part description on the complement of its
-natural sections, $Z$ is klt. The two natural sections of $pi$,
-called the axes, have divisor classes $xi-L_1$ and
-$xi-L_2$. The canonical-bundle formula for a projective bundle
+natural sections, Z is klt. The two natural sections of pi,
+called the axes, have divisor classes xi-L_1 and
+xi-L_2. The canonical-bundle formula for a projective bundle
gives
K_Zsim-((sum of the two axes)).
The complement of the two axes is called the torus open; a
-general point of $Z$ lies there.
+general point of Z lies there.
-The fibre of either projection $Zto X$ over a smooth point is, up
+The fibre of either projection Zto X over a smooth point is, up
to a constant one-dimensional factor,
[
Z_{sl}=PP(OO_X ⊕ OO_X(L)),
- qquad R_{sl}=P+qxi_{sl}.
+ R_{sl}=P+qxi_{sl}.
]
-Its axes have classes $xi_{sl}$ and
-$xi_{sl}-L$, and its canonical divisor is the negative of
+Its axes have classes xi_{sl} and
+xi_{sl}-L, and its canonical divisor is the negative of
their sum. The slice is projective and klt, so it satisfies the
singularity hypotheses needed to apply the base-component estimates
-of Section~sec:jet-tools, using a small $Q$-factorialization
+of Section~sec:jet-tools, using a small Q-factorialization
if necessary.
-Write $d=2n+1$. There are positive constants depending only on $n$
-such that, for each fixed $q$ and all sufficiently small $t$,
+Write d=2n+1. There are positive constants depending only on n
+such that, for each fixed q and all sufficiently small t,
c_nqε^{2n}le R^dle C_nqε^{2n},
- qquad
+
R_{sl}^{n+1}le C_{sl,n}qε^n.
-Here is the computation. Expanding $R^d=(P_1+P_2+qxi)^d$ in nef
+Here is the computation. Expanding R^d=(P_1+P_2+qxi)^d in nef
classes and using
[
- pi_*(xi^j)=sum_{a=0}^{j-1}L_1^aL_2^{j-1-a}qquad(jge1),
+ pi_*(xi^j)=sum_{a=0}^{j-1}L_1^aL_2^{j-1-a}(jge1),
]
-the term containing exactly one $xi$ is
-$(2n+1)binom{2n}{n}q(P^n)^2$, which gives the lower bound in
+the term containing exactly one xi is
+(2n+1)binom{2n}{n}q(P^n)^2, which gives the lower bound in
(j:bundle-volumes). For the upper bound, the mixed
-intersections of $L_i$ can be bounded by $P_i/r$ (since
-$(1)/(r)P_i=L_i+tA_i$, and $A_i$ can be absorbed by nonnegativity
-of the mixed-degree coefficients); after factoring out $q$, the
-remaining powers of $q/r$ are bounded by one once $rge q$, and
-there are only finitely many monomials depending on $n$. The slice
+intersections of L_i can be bounded by P_i/r (since
+(1)/(r)P_i=L_i+tA_i, and A_i can be absorbed by nonnegativity
+of the mixed-degree coefficients); after factoring out q, the
+remaining powers of q/r are bounded by one once rge q, and
+there are only finitely many monomials depending on n. The slice
calculation is identical, using
-$pi_*(xi^j)=L^{j-1}$.
+pi_*(xi^j)=L^{j-1}.
-For a nef rational Cartier divisor $R$ and a smooth point $z$, its
+For a nef rational Cartier divisor R and a smooth point z, its
Seshadri constant is
[
ε(R;z)=inf_{Cni z}(Rcdot C)/(mult_z C),
]
-where $C$ ranges over integral curves through $z$. A lower bound
-for this constant controls the degree of curves through $z$ relative
-to their multiplicity at $z$, which is what asymptotic Riemann--Roch
+where C ranges over integral curves through z. A lower bound
+for this constant controls the degree of curves through z relative
+to their multiplicity at z, which is what asymptotic Riemann--Roch
needs to make jets surjective.
-For a suitable fixed positive integer $q$, and all sufficiently
-small $t$ in (eq:scaling), there is a smooth point $z_*in Z$
+For a suitable fixed positive integer q, and all sufficiently
+small t in (eq:scaling), there is a smooth point z_*in Z
in the torus open, lying over smooth points of both factors of
-$Xtimes X$, such that
+Xtimes X, such that
ε(R;z_*)>2n+2.
-More generally, for any prescribed dense open subsets $U_1,U_2$ of
-the smooth locus of $X$, the point may be chosen over $U_1times U_2$.
+More generally, for any prescribed dense open subsets U_1,U_2 of
+the smooth locus of X, the point may be chosen over U_1times U_2.
In particular, these opens may be the isomorphism loci of fixed
-resolutions of $X$.
+resolutions of X.
## Proof of the large Seshadri constant
@@ -1507,78 +1493,78 @@ resolutions of $X$.
We prove the proposition in the remainder of this section.
Choices before the limit.: Choose a positive rational gap
-$delta$ such that
+delta such that
(2/delta)^nC_{sl,n}ε^n<1.
-Next choose $q$ so large that $d+1$ increasing levels of gap
-$delta$, with the lowest level strictly greater than $2n+2$, all lie
-strictly below $(c_nqε^{2n})^{1/d}$. Both $q,delta$ are
-fixed before $t$ tends to zero. Fix the prescribed opens
-$U_1,U_2$ if they are given.
+Next choose q so large that d+1 increasing levels of gap
+delta, with the lowest level strictly greater than 2n+2, all lie
+strictly below (c_nqε^{2n})^{1/d}. Both q,delta are
+fixed before t tends to zero. Fix the prescribed opens
+U_1,U_2 if they are given.
The reduction to a moving kernel.: Suppose
(j:large-seshadri) fails throughout the smooth torus preimage
-of $U_1times U_2$ for arbitrarily small $t$. Fix such a $t$.
+of U_1times U_2 for arbitrarily small t. Fix such a t.
Asymptotic Riemann--Roch and the elementary jet count at a smooth
point show that all the full kernels at the chosen levels are
-nonzero for sufficiently large divisible $k$, because the highest
-level has $d$-th power smaller than $R^d$. More explicitly: the
-length of the quotient by jets of order $<ktau$ is
-$binom{ktau+d}{d}sim(ktau)^d/d!$, and the number of sections is
-$chi(OO(kR))sim(kR)^d/d!=(ktau)^d/d!cdot (R/tau)^d$, which
-dominates as long as $tau^d<R^d$. For our chosen levels
-$tau_i<(c_nqε^{2n})^{1/d}le (R^d)^{1/d}$, this holds.
+nonzero for sufficiently large divisible k, because the highest
+level has d-th power smaller than R^d. More explicitly: the
+length of the quotient by jets of order <ktau is
+binom{ktau+d}{d}sim(ktau)^d/d!, and the number of sections is
+chi(OO(kR))sim(kR)^d/d!=(ktau)^d/d!cdot (R/tau)^d, which
+dominates as long as tau^d<R^d. For our chosen levels
+tau_i<(c_nqε^{2n})^{1/d}le (R^d)^{1/d}, this holds.
Failure of (j:large-seshadri) supplies, through each very
-general point, a curve $C$ with $Rcdot C/mult_zC$ smaller than the
-lowest level $tau_0$. Every section in the lowest kernel vanishes
-identically on $C$: otherwise its intersection with $C$ would be at
-least its vanishing order times $mult_zC$, contradicting the degree
-inequality $Rcdot C<tau_0mult_zC$ and the order bound $ktau_0$.
+general point, a curve C with Rcdot C/mult_zC smaller than the
+lowest level tau_0. Every section in the lowest kernel vanishes
+identically on C: otherwise its intersection with C would be at
+least its vanishing order times mult_zC, contradicting the degree
+inequality Rcdot C<tau_0mult_zC and the order bound ktau_0.
Thus the lowest base locus has a positive-dimensional component
through the marked point.
-Take $k$ divisible enough that $kdeltage4$ and apply
+Take k divisible enough that kdeltage4 and apply
Lemma~j:moving-kernel. We obtain a family of proper integral
-components $Vsubset Z$ whose incidence dominates $Z$. For general
-members of dimension $f$, the estimates are
+components Vsubset Z whose incidence dominates Z. For general
+members of dimension f, the estimates are
0<R^fcdot V&le(2/delta)^{d-f}R^d,
K_{V^*}R^{f-1}&le(K_Z+cR)|_Vcdot R^{f-1},
- qquad 0<cle 2(d-f)/delta.
+ 0<cle 2(d-f)/delta.
Here and below, classes on a resolution are understood by pullback.
-The positivity $R^fcdot V>0$ follows from ampleness of $R$. General
+The positivity R^fcdot V>0 follows from ampleness of R. General
members meet the torus open, so (j:axes-canonical) makes the
ambient canonical contribution nonpositive. All constants in these
-bounds are uniform once $q,ε,delta$ have been fixed.
+bounds are uniform once q,ε,delta have been fixed.
-The remainder of the proof rules out all possible $(V,f)$. We first
-constrain the two projections of $V$ to $X$.
+The remainder of the proof rules out all possible (V,f). We first
+constrain the two projections of V to X.
## Excluding positive-dimensional slice fibres
-Consider either projection $Vto X$. Let $F$ be the component of a
-general fibre of $Vtoim(Vto X)$ through a general
-incidence point of $V$. Choose the incidence point in the opens
+Consider either projection Vto X. Let F be the component of a
+general fibre of Vtoim(Vto X) through a general
+incidence point of V. Choose the incidence point in the opens
given by Lemma~j:fiber-restriction, over very general points of
-both factors of $X$, and in the smooth torus open. We claim
+both factors of X, and in the smooth torus open. We claim
dim Fnotin{1,...,n}.
-Suppose, toward a contradiction, that $1le f_{sl}=dim F
-le n$. The ambient fibre of either projection $Zto X$ over a
-smooth point is the slice $Z_{sl}$ of dimension $n+1$.
+Suppose, toward a contradiction, that 1le f_{sl}=dim F
+le n. The ambient fibre of either projection Zto X over a
+smooth point is the slice Z_{sl} of dimension n+1.
Lemma~j:fiber-restriction gives a nonzero restricted subseries
-with $F$ an isolated base component, whose base ideal is contained in
-$mathcal I_F^h$calI_F^h$ with $h=lceil kdelta/2rceil$. Generic smoothness
-is imposed over the actual image of $V$, which is why a general fibre
+with F an isolated base component, whose base ideal is contained in
+mathcal I_F^hcalI_F^h with h=lceil kdelta/2rceil. Generic smoothness
+is imposed over the actual image of V, which is why a general fibre
through the incidence point was chosen; the chosen point is smooth
in the ambient slice by the very general choice. Applying
Lemma~j:base-component on that slice gives
@@ -1592,37 +1578,37 @@ Lemma~j:base-component on that slice gives
R_{sl}^{f_{sl}-1},
-with $cle 2(n+1-f_{sl})/delta$. Cartier multiples are
+with cle 2(n+1-f_{sl})/delta. Cartier multiples are
chosen before restriction, so the constant line from the fixed
factor causes no change in this calculation.
-Let $W_0subset X$ be the image of $F$ in the other factor. Three
+Let W_0subset X be the image of F in the other factor. Three
cases arise.
-Case 1: $F$ is generically finite over a proper
-subvariety.: Suppose $Fto W_0$ is generically finite with
-$dim W_0<n$. Let $P_Y=R_{sl}|_{F^*}$, and let $L_Y$ be the
-pullback of $L$ from $X$. The first inequality in
+Case 1: F is generically finite over a proper
+subvariety.: Suppose Fto W_0 is generically finite with
+dim W_0<n. Let P_Y=R_{sl}|_{F^*}, and let L_Y be the
+pullback of L from X. The first inequality in
(j:slice-bounds) gives a uniform positive volume bound; since
-$F$ meets neither axis generically, $K_{Z_{sl}}|_Fle0$, so
+F meets neither axis generically, K_{Z_{sl}}|_Fle0, so
the second inequality gives a uniform canonical ratio
-$K_{F^*}P_Y^{f_{sl}-1}le cP_Y^{f_{sl}}$. Moreover
-$P_Y-rL_Y$ is nef, since
-$P_Y-rL_Y=(P|_{W_0}-rL|_{W_0})+qxi_{sl}|_{F^*}$ is the sum
-of two nef classes. The image $W_0$ contains the very general
+K_{F^*}P_Y^{f_{sl}-1}le cP_Y^{f_{sl}}. Moreover
+P_Y-rL_Y is nef, since
+P_Y-rL_Y=(P|_{W_0}-rL|_{W_0})+qxi_{sl}|_{F^*} is the sum
+of two nef classes. The image W_0 contains the very general
projection of the incidence point, so (j:lower-big) applies and
-gives bigness of $K_{F^*}+L_Y$. Lemma~lemma:bounded-curves
-excludes this case for large $r$.
+gives bigness of K_{F^*}+L_Y. Lemma~lemma:bounded-curves
+excludes this case for large r.
-Case 2: $F$ is a projective-line bundle over a proper
-subvariety.: Suppose $Fto W_0$ is not generically finite. Its
+Case 2: F is a projective-line bundle over a proper
+subvariety.: Suppose Fto W_0 is not generically finite. Its
general fibre has dimension one, and since the ambient projection
-$Z_{sl}to X$ is a $PP^1$-bundle, the integral $F$ must
-coincide with the full $PP^1$-bundle over $W_0$. Write
-$w=dim W_0=f_{sl}-1$.
+Z_{sl}to X is a PP^1-bundle, the integral F must
+coincide with the full PP^1-bundle over W_0. Write
+w=dim W_0=f_{sl}-1.
-If $w=0$, then $F$ is a projective-line fibre and
-$R_{sl}cdot F=q$. The first inequality in
+If w=0, then F is a projective-line fibre and
+R_{sl}cdot F=q. The first inequality in
(j:slice-bounds), together with (j:bundle-volumes) and
(j:slice-gap), gives
[
@@ -1631,81 +1617,81 @@ $R_{sl}cdot F=q$. The first inequality in
]
impossible.
-Suppose $w>0$. Resolve $W_0$ to $W_0^*$, and take as a resolution
-of $F$ the induced projective bundle $p_0colon F^*to W_0^*$. Its
-relative canonical divisor is $-2xi+p_0^*L$, exactly the
-restriction of $K_{Z_{sl}}$. Cancelling this term in
+Suppose w>0. Resolve W_0 to W_0^*, and take as a resolution
+of F the induced projective bundle p_0colon F^*to W_0^*. Its
+relative canonical divisor is -2xi+p_0^*L, exactly the
+restriction of K_{Z_{sl}}. Cancelling this term in
(j:slice-bounds) gives
p_0^*K_{W_0^*}R_{sl}^w
le cR_{sl}^{w+1}cdot F.
-The divisor $K_{W_0^*}$ is pseudo-effective by the very general point
+The divisor K_{W_0^*} is pseudo-effective by the very general point
condition. On expanding the left side of (j:saturated-canonical)
-and pushing down, its term with exactly one $xi$ is
-$wqK_{W_0^*}P^{w-1}$; every other nonzero term is nonnegative, since
-it pairs the pseudo-effective $K_{W_0^*}$ with nef classes. On the
+and pushing down, its term with exactly one xi is
+wqK_{W_0^*}P^{w-1}; every other nonzero term is nonnegative, since
+it pairs the pseudo-effective K_{W_0^*} with nef classes. On the
right, the expansion used for (j:bundle-volumes) gives
(w+1)qP^wcdot W_0
le R_{sl}^{w+1}cdot F
le C'_n qP^wcdot W_0,
-using $rge q$. The leading coefficient in the lower bound is the
-term with exactly one $xi$, as before. The first inequality in
-(j:slice-bounds) then bounds $0<P^wcdot W_0$ uniformly from
+using rge q. The leading coefficient in the lower bound is the
+term with exactly one xi, as before. The first inequality in
+(j:slice-bounds) then bounds 0<P^wcdot W_0 uniformly from
above, and (j:saturated-volume)'s leading coefficient bounds it
from below. Equations (j:saturated-canonical)--(j:saturated-volume)
together give
[
K_{W_0^*}P^{w-1}le(cC'_n)/(w)P^wcdot W_0.
]
-Now apply (j:lower-big) on $W_0^*$, with the pullback of $P$ as
-the testing class and $L$ as the full-nef-dimension divisor.
-$W_0$ contains a very general point of $X$, so the required bigness
+Now apply (j:lower-big) on W_0^*, with the pullback of P as
+the testing class and L as the full-nef-dimension divisor.
+W_0 contains a very general point of X, so the required bigness
and full nef dimension hold by Lemma~j:very-general-images.
-Lemma~lemma:bounded-curves excludes this case for large $r$.
-
-Case 3: $F$ is a multisection over $X$.: The only
-remaining possibility is that $f_{sl}=n$ and $Fto X$ is
-generically finite of degree $e>0$. Intersect $F$ with each of the
-two Cartier axes and push the resulting cycles down to $X$. This
-gives effective integral Weil divisors $D_1,D_2$. The axes do not
-contain $F$, and their classes differ by $L$; taking the norm of the
+Lemma~lemma:bounded-curves excludes this case for large r.
+
+Case 3: F is a multisection over X.: The only
+remaining possibility is that f_{sl}=n and Fto X is
+generically finite of degree e>0. Intersect F with each of the
+two Cartier axes and push the resulting cycles down to X. This
+gives effective integral Weil divisors D_1,D_2. The axes do not
+contain F, and their classes differ by L; taking the norm of the
associated rational section, or equivalently its projection, gives
D_1-D_2simpm eL.
-For either axis $S$, the classes $R_{sl}-qS$ and
-$R_{sl}-P$ are nef. Intersecting with the effective cycle
-$Scdot F$ and then using nefness on $F$ yields
+For either axis S, the classes R_{sl}-qS and
+R_{sl}-P are nef. Intersecting with the effective cycle
+Scdot F and then using nefness on F yields
qD_iP^{n-1}le qScdot Fcdot R_{sl}^{n-1}
le R_{sl}^ncdot F.
Thus both axis-image degrees are uniformly bounded.
-Set $G=D_1+D_2$. It is $Q$-Cartier because $X$ is $Q$-factorial.
-Choose $b>0$ rational sufficiently small that $(X,bG)$ is klt. We
+Set G=D_1+D_2. It is Q-Cartier because X is Q-factorial.
+Choose b>0 rational sufficiently small that (X,bG) is klt. We
claim
kappa(K_X+bG)>0.
-The Iitaka dimension is at least zero, since $K_X+bGsim_Q bG$.
+The Iitaka dimension is at least zero, since K_X+bGsim_Q bG.
If it were zero, Theorems~input:models and~input:abundance
-would give an ordinary MMP ending at a good model $Y$ whose
+would give an ordinary MMP ending at a good model Y whose
transformed adjoint is rationally linearly trivial. The transform of
-$K_X$ is then linearly trivial as well, so the effective transform
-of $G$ must vanish: every component of $G$ has been contracted.
-On a common resolution $acolon Uto X$ and $b_Ycolon Uto Y$,
-relation (j:signed-L) then expresses $a^*L$ up to rational
-linear equivalence as a signed $b_Y$-exceptional divisor $E$.
-Divisors exceptional over $X$ remain exceptional over $Y$, because
-the MMP extracts no divisors. Nefness of $a^*L$ and the negativity
-lemma give $Ele0$. Its class is also pseudo-effective.
+K_X is then linearly trivial as well, so the effective transform
+of G must vanish: every component of G has been contracted.
+On a common resolution acolon Uto X and b_Ycolon Uto Y,
+relation (j:signed-L) then expresses a^*L up to rational
+linear equivalence as a signed b_Y-exceptional divisor E.
+Divisors exceptional over X remain exceptional over Y, because
+the MMP extracts no divisors. Nefness of a^*L and the negativity
+lemma give Ele0. Its class is also pseudo-effective.
A nonzero anti-effective divisor has negative intersection with a
-sufficiently high power of an ample divisor, so $E=0$. This
-contradicts full nef dimension of $L$, and thus proves
+sufficiently high power of an ample divisor, so E=0. This
+contradicts full nef dimension of L, and thus proves
(j:positive-boundary-kappa).
The already proved positive-Iitaka-dimension case
@@ -1713,87 +1699,87 @@ The already proved positive-Iitaka-dimension case
[
K_X+bG+L big.
]
-Its full-positivity hypothesis holds because $L$ has full nef
-dimension and $K_X+bG$ is pseudo-effective.
+Its full-positivity hypothesis holds because L has full nef
+dimension and K_X+bG is pseudo-effective.
-Take a log resolution $acolon Yto X$ of $G$, and let $D_0$ be the
+Take a log resolution acolon Yto X of G, and let D_0 be the
reduced simple normal crossing divisor consisting of the strict
-support of $G$ together with every exceptional divisor. Decrease $b$
+support of G together with every exceptional divisor. Decrease b
if necessary; Proposition~prop:full-positive still applies.
-Terminality of $X$ and the support of $a^*G$ then give
+Terminality of X and the support of a^*G then give
[
K_Y+D_0+a^*L ge a^*(K_X+bG+L)
]
up to rational linear equivalence and an effective difference.
Indeed the finitely many exceptional discrepancies are positive, and
-decreasing $b$ makes each coefficient of $b,a^*G$ no larger than the
-corresponding coefficient of $K_Y-a^*K_X+D_0$. Hence the left side
+decreasing b makes each coefficient of b,a^*G no larger than the
+corresponding coefficient of K_Y-a^*K_X+D_0. Hence the left side
is big.
-Test on $Y$ with $a^*P$. Exceptional divisors contribute zero to
-intersection with $(a^*P)^{n-1}$, and $K_Xsim0$. The degree of the
-reduced strict support is no larger than $GP^{n-1}$, which is bounded
-by (j:axis-degrees); also $LP^{n-1}le P^n/r$. Since
-(j:volume-window) bounds $P^n$ away from zero, we obtain a
-uniform constant $C$ such that
+Test on Y with a^*P. Exceptional divisors contribute zero to
+intersection with (a^*P)^{n-1}, and K_Xsim0. The degree of the
+reduced strict support is no larger than GP^{n-1}, which is bounded
+by (j:axis-degrees); also LP^{n-1}le P^n/r. Since
+(j:volume-window) bounds P^n away from zero, we obtain a
+uniform constant C such that
[
(K_Y+D_0+a^*L)(a^*P)^{n-1}le C(a^*P)^n.
]
Lemma~lemma:bounded-curves, now with its reduced boundary
-$D_0$, gives the final contradiction. This proves
+D_0, gives the final contradiction. This proves
(j:no-intermediate-fibre).
## Dimension reduction for moving components
-Thus a general moving component $V$ has fibre dimension either zero
-or $n+1$ over its image in either factor. Fibre dimension $n+1$
-would make $V$ the full inverse image of its image in that factor.
-Since $V$ is proper, the image has dimension at most $n-1$, so the
-other projection would have fibre dimension between $1$ and $n$,
+Thus a general moving component V has fibre dimension either zero
+or n+1 over its image in either factor. Fibre dimension n+1
+would make V the full inverse image of its image in that factor.
+Since V is proper, the image has dimension at most n-1, so the
+other projection would have fibre dimension between 1 and n,
already excluded by (j:no-intermediate-fibre). Both
-projections of $V$ are therefore generically finite onto their
-images, and $dim Vle n$. If $dim V<n$, the bounds
-(j:total-component-bounds), the nef class $R-rL_i$, and
+projections of V are therefore generically finite onto their
+images, and dim Vle n. If dim V<n, the bounds
+(j:total-component-bounds), the nef class R-rL_i, and
(j:lower-big) again contradict Lemma~lemma:bounded-curves.
-Only $n$-dimensional correspondences, dominant over both factors of
-$Xtimes X$, remain for further analysis.
+Only n-dimensional correspondences, dominant over both factors of
+Xtimes X, remain for further analysis.
## Correspondences and their branch divisors
-For the remaining $V$: dimension $n$, dominant over both factors of
-$Xtimes X$. Equation~(j:total-component-bounds) gives uniform
+For the remaining V: dimension n, dominant over both factors of
+Xtimes X. Equation~(j:total-component-bounds) gives uniform
bounds
R^ncdot Vle C,
- qquad
+
K_{V^*}R^{n-1}le cR^ncdot Vle C,
-after increasing $C$. Each projection has bounded degree:
-$R-P_i$ is nef, and $P^n$ is bounded below.
+after increasing C. Each projection has bounded degree:
+R-P_i is nef, and P^n is bounded below.
-For each fixed $t$, resolve the general members in their algebraic
+For each fixed t, resolve the general members in their algebraic
family and shrink the parameter space so that the resolutions form a
-smooth projective family with integral fibres and morphisms to $X$ in
+smooth projective family with integral fibres and morphisms to X in
both slots. The dominant incidence and all numerical bounds
persist. Finite extensions of the parameter space are allowed.
-Over $X_{sm}$, name the components of the relative Jacobian
+Over X_{sm}, name the components of the relative Jacobian
divisors as in Lemma~j:finite-covers. Consider a component
-whose image on a general member is a divisor $Jsubset X$.
-
-Such divisors $J$ cannot sweep $X$. If they did, choose a general
-one through a very general point. Let $E_J$ be a ramification
-divisor over it on the corresponding $V^*$. Pullback of a
-canonical form from $X$ gives an effective canonical divisor on
-$V^*$. Regularity follows by factoring through a resolution of $X$
-and using its canonical singularities; the coefficient of $E_J$ is
-positive because its image meets $X_{sm}$ generically. The
-difference $R-P_i$ is nef, so (j:correspondence-bounds) and the
+whose image on a general member is a divisor Jsubset X.
+
+Such divisors J cannot sweep X. If they did, choose a general
+one through a very general point. Let E_J be a ramification
+divisor over it on the corresponding V^*. Pullback of a
+canonical form from X gives an effective canonical divisor on
+V^*. Regularity follows by factoring through a resolution of X
+and using its canonical singularities; the coefficient of E_J is
+positive because its image meets X_{sm} generically. The
+difference R-P_i is nef, so (j:correspondence-bounds) and the
projection formula bound
[
0<d_J:=P^{n-1}cdot Jle C.
]
-On a resolution $J^*$, Lemma~j:subadjunction, applied to
-$(X,J)$ at the generic point of $J$, gives
+On a resolution J^*, Lemma~j:subadjunction, applied to
+(X,J) at the generic point of J, gives
K_{J^*}P^{n-2}
@@ -1802,50 +1788,50 @@ $(X,J)$ at the generic point of $J$, gives
lefrac{(JP^{n-1})^2}{P^n}le C' d_J.
-The pair is lc at that generic point because $X$ is normal and hence
+The pair is lc at that generic point because X is normal and hence
smooth there. The middle inequality is the Hodge index inequality
on a resolution, obtained by ample approximation of the pulled-back
-testing class; it does not require $J$ to be nef. The final constant
-is uniform by the volume lower bound and the bound for $d_J$.
+testing class; it does not require J to be nef. The final constant
+is uniform by the volume lower bound and the bound for d_J.
-Since $J$ sweeps $X$, its resolution satisfies the very general point
+Since J sweeps X, its resolution satisfies the very general point
conditions used in (j:lower-big). With testing class
-$P|_{J^*}$, whose top self-intersection is $d_J$,
+P|_{J^*}, whose top self-intersection is d_J,
Equation~(j:branch-canonical) gives precisely the hypotheses
-of Lemma~lemma:bounded-curves in dimension $n-1$. This
+of Lemma~lemma:bounded-curves in dimension n-1. This
contradiction excludes sweeping branch divisors.
Lemma~j:finite-covers now gives, for each slot and this fixed
-$t$, only finitely many possible finite normal Stein covers of $X$,
-up to isomorphism over $X$. The point of the preceding step is that
+t, only finitely many possible finite normal Stein covers of X,
+up to isomorphism over X. The point of the preceding step is that
degree bounds alone would not give this conclusion: non-sweeping
branch images provide a fixed smooth branch complement for the
family. Neither that complement nor the resulting finite lists need
-be uniform in $t$.
+be uniform in t.
## Fixed covers and an abelian model
-Each $V^*$ maps birationally to both of its normal Stein covers, and
+Each V^* maps birationally to both of its normal Stein covers, and
hence determines a graph of a birational map between a pair of
covers drawn from the two finite lists. We explain why some fixed
pair has a family of graphs dominating its product.
Restrict the finite set of pairs to those realized by at least one
-original member $V^*$ satisfying (j:correspondence-bounds). The
-original incidence dominates $Z$, so the projected incidences for
-these pairs cover a dense open of $Xtimes X$. For a fixed pair,
+original member V^* satisfying (j:correspondence-bounds). The
+original incidence dominates Z, so the projected incidences for
+these pairs cover a dense open of Xtimes X. For a fixed pair,
graphs of birational maps lie in countably many Hilbert schemes.
Stratify these schemes so that the fibres are integral and flat and
both projections are birational; the latter conditions are tested by
-dominance and degree one after shrinking. Because $C$ is
-uncountable, a dense open of $Xtimes X$ cannot be covered by
+dominance and degree one after shrinking. Because C is
+uncountable, a dense open of Xtimes X cannot be covered by
countably many proper closed subsets. Thus, for one fixed pair
-$T_1,T_2$, some integral graph family has dominant evaluation to
-$Xtimes X$. Since $T_1times T_2$ is finite over $Xtimes X$ and
+T_1,T_2, some integral graph family has dominant evaluation to
+Xtimes X. Since T_1times T_2 is finite over Xtimes X and
has the same dimension, this evaluation is dominant onto
-$T_1times T_2$ as well.
+T_1times T_2 as well.
-Both covers are non-uniruled because they are finite over $X$. The
+Both covers are non-uniruled because they are finite over X. The
birational-group theorem of Hanamura
cite[Theorems~2.1--2.2]{Hanamura}, in the form recalled in
cite[Proposition~3.7 and Theorems~3.8--3.9]{Blanc}, implies that
@@ -1868,67 +1854,67 @@ canonical nonvanishing conclusion from that proposition is being
used.
This structural conclusion depends only on the two covers. We may
-therefore return to an original member $V^*$ of this pair, which
+therefore return to an original member V^* of this pair, which
retains (j:correspondence-bounds), and resolve further to
obtain a birational morphism
[
p_0colon V^*→ A_0
]
to an abelian variety. Higher resolutions do not change the
-canonical intersection with the pulled-back $(n-1)$-st power of $R$.
-Let $L_0$ be the pullback of $L$ from either slot. It is nef,
+canonical intersection with the pulled-back (n-1)-st power of R.
+Let L_0 be the pullback of L from either slot. It is nef,
Cartier, and of full nef dimension. We claim
K_{V^*}+L_0 is big.
-The pushforward $N_0=(p_0)_*L_0$ is pseudo-effective on $A_0$, and
+The pushforward N_0=(p_0)_*L_0 is pseudo-effective on A_0, and
is therefore nef; on an abelian variety, translations make the
-pseudo-effective and nef cones coincide. If $N_0$ were not big, the
+pseudo-effective and nef cones coincide. If N_0 were not big, the
standard description of nef line bundles on abelian varieties would
give a positive-dimensional abelian subvariety on whose translates
-$N_0$ is numerically trivial. This follows from the semipositive
+N_0 is numerically trivial. This follows from the semipositive
Hermitian form of the numerical class: its kernel is rational for the
lattice, since the alternating form of the integral Cartier class is
integral, and therefore defines an abelian subvariety. A general
-translate meets the image of the exceptional locus of $p_0$ in
+translate meets the image of the exceptional locus of p_0 in
codimension at least two, if at all. Complete-intersection curves
in that translate can thus be chosen through very general points and
-avoiding this image. Their lifts have $L_0$-degree zero,
-contradicting full nef dimension. Hence $N_0$ is big.
+avoiding this image. Their lifts have L_0-degree zero,
+contradicting full nef dimension. Hence N_0 is big.
By the negativity lemma,
[
p_0^*N_0-L_0=E_0ge0
]
-is exceptional. The canonical divisor of $V^*$ has an effective
-representative $E_K$ with positive coefficient along every
-$p_0$-exceptional divisor, because the target is smooth with trivial
-canonical bundle. Choose a rational $0<u<1$ sufficiently small that
-$E_K-uE_0ge0$. Then
+is exceptional. The canonical divisor of V^* has an effective
+representative E_K with positive coefficient along every
+p_0-exceptional divisor, because the target is smooth with trivial
+canonical bundle. Choose a rational 0<u<1 sufficiently small that
+E_K-uE_0ge0. Then
[
K_{V^*}+uL_0sim_Q u p_0^*N_0+(E_K-uE_0)
]
-is big. Adding the nef divisor $(1-u)L_0$ proves
+is big. Adding the nef divisor (1-u)L_0 proves
(j:abelian-big).
-Now apply Lemma~lemma:bounded-curves in dimension $n$ on $V^*$,
-with testing class $R$ and empty boundary. The bounds
+Now apply Lemma~lemma:bounded-curves in dimension n on V^*,
+with testing class R and empty boundary. The bounds
(j:correspondence-bounds) give both the top intersection bound
-and the canonical ratio, while $R-rL_0$ is nef and
+and the canonical ratio, while R-rL_0 is nef and
(j:abelian-big) supplies bigness. This final contradiction
excludes the correspondence case.
-subsection{Conclusion of the proof of Proposition~prop:large-seshadri}
+Conclusion of the proof of Proposition~prop:large-seshadri
The alternatives for every moving base component have been
exhausted: first its slice fibres, then components of dimension less
-than $n$, then sweeping branch divisors, and finally the fixed
+than n, then sweeping branch divisors, and finally the fixed
covers. In each case the contradiction comes from
-Lemma~lemma:bounded-curves with constants uniform in $t$. The
-finite-cover lists were used only for one fixed $t$, and their sizes
+Lemma~lemma:bounded-curves with constants uniform in t. The
+finite-cover lists were used only for one fixed t, and their sizes
do not enter the degree bounds. Thus failures of
-(j:large-seshadri) cannot occur along a sequence $tto0$. This
+(j:large-seshadri) cannot occur along a sequence tto0. This
proves the proposition, including the additional open conditions on
the chosen point.
@@ -1936,7 +1922,7 @@ We have now constructed, under the hypothetical failure of
Theorem~thm:terminal-big, the scalar estimate of
Lemma~lemma:scalar-bound and the projective-bundle estimate of
Proposition~prop:large-seshadri. Section~sec:frobenius
-fixes one sufficiently small value of $t$ and places these data in
+fixes one sufficiently small value of t and places these data in
the finite-data Frobenius comparison.
# Frobenius incompatibility
@@ -1947,7 +1933,7 @@ projective-bundle estimate of Proposition~prop:large-seshadri
are incompatible. Their incompatibility is expressed by the following
theorem about fixed complex data, whose proof occupies this section.
The reduction to positive characteristic is managed so that every
-constant is uniform in the residual characteristic $pto∞$.
+constant is uniform in the residual characteristic pto∞.
The shape of the argument is as follows. Two orders of vanishing are
compared for one determinant. Ordinary jets make the determinant
@@ -1960,57 +1946,57 @@ gives the opposite inequality.
## The finite-data statement
For a vector bundle on a smooth projective curve, write
-$mu_{min}$ and $mu_{max}$ for its minimum and maximum
+mu_{min} and mu_{max} for its minimum and maximum
Harder--Narasimhan slopes, i.e. degrees divided by ranks of the
extremal sub- or quotient bundles. For a line bundle on a smooth
-variety, generate jets through order $a$ at a point means the
+variety, generate jets through order a at a point means the
evaluation map from sections to the quotient of the local ring by the
-$(a+1)$-st power of the maximal ideal is surjective.
+(a+1)-st power of the maximal ideal is surjective.
[Finite-data Frobenius incompatibility]
-Let $W$ be a smooth connected projective complex variety of dimension
-$nge2$. Let $D,H$ be rational Cartier divisors with $H$ ample, let
-$S$ be an integral Cartier divisor, and fix an integer $q>0$ and real
-numbers $r>1$, $ε>0$. Suppose
+Let W be a smooth connected projective complex variety of dimension
+nge2. Let D,H be rational Cartier divisors with H ample, let
+S be an integral Cartier divisor, and fix an integer q>0 and real
+numbers r>1, ε>0. Suppose
- H^nle(2ε)^n,qquad
- K_WH^{n-1}le 2 H^n/r,qquad
+ H^nle(2ε)^n,
+ K_WH^{n-1}le 2 H^n/r,
(2D+qS)H^{n-1}le 4H^n,
- (14ε)^n<tfrac14,qquad 80ε<tfrac34.
+ (14ε)^n<tfrac14, 80ε<tfrac34.
Assume that the following fixed data exist.
renewcommand{labelenumi}{(roman{enumi)}}
-item A smooth integral complete-intersection flag from $W$ to
-a curve $C$, whose successive divisor classes are $h_i=l_iH$ for
-fixed positive integers $l_i$ for which $l_iH$ is Cartier, with
-$mu_{min}(Ω_W^1|_C)ge0$.
-item On the projective bundle
+ A smooth integral complete-intersection flag from W to
+a curve C, whose successive divisor classes are h_i=l_iH for
+fixed positive integers l_i for which l_iH is Cartier, with
+mu_{min}(Ω_W^1|_C)ge0.
+ On the projective bundle
[
Z=PP(mathcal O_W(S)_1calO_W(S)_1 ⊕ mathcal O_W(S)_2)calO_W(S)_2)
→ Wtimes W,
- qquad xi=c_1(mathcal O_Z(1)),calO_Z(1)),
- qquad M=D_1+D_2+qxi,
+ xi=c_1(mathcal O_Z(1)),calO_Z(1)),
+ M=D_1+D_2+qxi,
]
-use the convention that the direct image of $mathcal O_Z(acalO_Z(axi)$ is
-the $a$-th symmetric power of the displayed sum for $age0$; subscripts
-indicate pullback from the corresponding factor of $Wtimes W$. The
+use the convention that the direct image of mathcal O_Z(acalO_Z(axi) is
+the a-th symmetric power of the displayed sum for age0; subscripts
+indicate pullback from the corresponding factor of Wtimes W. The
two axes are the sections defined by the two summands. A smooth point
-$z_*$ in their complement, together with finitely many sections of
-$k_0M$ for some integer $k_0>0$ with $k_0D$ Cartier, generate jets
-through order $(2n+2)k_0$ at $z_*$.
-item A point $xin W$, its blowup
-$pi_xcolonwidehatwidehat WWto W$ with exceptional divisor $J$, and a
+z_* in their complement, together with finitely many sections of
+k_0M for some integer k_0>0 with k_0D Cartier, generate jets
+through order (2n+2)k_0 at z_*.
+ A point xin W, its blowup
+pi_xcolonwidehatwidehat WWto W with exceptional divisor J, and a
curve class
[
gamma=f_*(A_1... A_{n-1}),
]
-where $fcolon Vtowidehatwidehat W$W$ is one fixed birational morphism,
-$V$ is smooth integral projective, and the $A_i$ are ample integral
-Cartier divisors, such that $Jcdotgamma>0$ and
+where fcolon Vtowidehatwidehat WW is one fixed birational morphism,
+V is smooth integral projective, and the A_i are ample integral
+Cartier divisors, such that Jcdotgamma>0 and
pi_x^*(D+K_W/2+lambda S)cdotgamma
le 40ε, Jcdotgammaquad(0lelambdale q).
@@ -2020,9 +2006,9 @@ It suffices to check the two endpoints in this affine inequality.
These data cannot coexist.
-Note that $x$ is independent of the two base coordinates of $z_*$.
-No nefness is required of $D$, $S$, or $K_W$, and the theorem does
-not assume $K_W$ pseudo-effective. Its hypotheses specify the flag
+Note that x is independent of the two base coordinates of z_*.
+No nefness is required of D, S, or K_W, and the theorem does
+not assume K_W pseudo-effective. Its hypotheses specify the flag
and the movable curve class themselves. All these data will be
fixed before the residual characteristic tends to infinity.
@@ -2034,20 +2020,20 @@ hypotheses to finitely generated models and converting the fixed
ordinary jets into jets of order proportional to the residual
characteristic.
-Put $Λ=prod_{i=1}^{n-1}l_i$. The complete-intersection flag
+Put Λ=prod_{i=1}^{n-1}l_i. The complete-intersection flag
satisfies
[C]=Λ H^{n-1},
- qquad h_icdot C=l_iΛ H^n.
+ h_icdot C=l_iΛ H^n.
-Choose one sufficiently ample integral Cartier divisor $T_0$ so that
-every $T_0+aS$, for integers $0le ale(k_0-1)q$, has a section
-nonvanishing at each of the two base coordinates of $z_*$. Fix these
+Choose one sufficiently ample integral Cartier divisor T_0 so that
+every T_0+aS, for integers 0le ale(k_0-1)q, has a section
+nonvanishing at each of the two base coordinates of z_*. Fix these
finitely many sections.
All models, morphisms, divisors, points, the flag, the
-complete-intersection witness for $gamma$, and the jet and filler
-sections spread over an integral finitely generated $Z$-algebra of
+complete-intersection witness for gamma, and the jet and filler
+sections spread over an integral finitely generated Z-algebra of
characteristic zero. Shrink its spectrum so that:
the fibres and the flag are smooth projective and geometrically
integral; birational morphisms remain birational; the fixed ample
@@ -2056,13 +2042,13 @@ conditions persist. Birationality is preserved by spreading an
isomorphism between dense opens. All displayed intersection numbers
are constant on the spread.
-We also preserve $mu_{min}(Ω_W^1|_C)ge 0$. One route is to
+We also preserve mu_{min}(Ω_W^1|_C)ge 0. One route is to
spread the characteristic-zero Harder--Narasimhan filtration, make its
factors locally free, and use openness of their semistability on
curves; the degrees and ranks are fixed. Equivalently, a single
relative ample twist globally generates the curve bundle. Every
-quotient of rank $a$ then has degree bounded below by $-ad$ for a
-fixed integer $d$. A negative-degree quotient has one of finitely
+quotient of rank a then has degree bounded below by -ad for a
+fixed integer d. A negative-degree quotient has one of finitely
many ranks and degrees. The proper relative Quot schemes for those
Hilbert polynomials have images missing the generic point: after
removal of torsion, a negative-degree coherent quotient there would
@@ -2072,7 +2058,7 @@ finitely many images proves openness directly.
The base has closed points in arbitrarily large characteristics.
Take algebraic closures of their residue fields and exclude the
primes that divide relevant denominators. Retain notation for the
-reductions. The curve $gamma$ continues to pair nonnegatively with
+reductions. The curve gamma continues to pair nonnegatively with
every effective divisor: pull that divisor back under the fixed
birational morphism and intersect with the fixed ample divisors.
This tests effective divisors on the reduction itself. No
@@ -2081,47 +2067,47 @@ specialization of the characteristic-zero effective cone is used.
## Positive-characteristic jets
-For a sufficiently large residual characteristic $p$, set
+For a sufficiently large residual characteristic p, set
[
N_p=lfloor p/k_0rfloor,
- qquad B_p=k_0N_pD+T_0.
+ B_p=k_0N_pD+T_0.
]
-Then the sections of $pqxi+B_{p,1}+B_{p,2}$ generate jets through
-order $(2n+2)k_0N_p$ at $z_*$. In particular, they surject onto the
-quotient of the local ring by the $p$-th powers of all regular
+Then the sections of pqxi+B_{p,1}+B_{p,2} generate jets through
+order (2n+2)k_0N_p at z_*. In particular, they surject onto the
+quotient of the local ring by the p-th powers of all regular
parameters.
Proof.
-Multiply $N_p$ copies of the fixed jet system. Every monomial of
-total degree at most $(2n+2)k_0N_p$ is a product of $N_p$ monomials
-of degree at most $(2n+2)k_0$. Taking sections with those leading
+Multiply N_p copies of the fixed jet system. Every monomial of
+total degree at most (2n+2)k_0N_p is a product of N_p monomials
+of degree at most (2n+2)k_0. Taking sections with those leading
monomials gives a triangular system, ordered by total degree. Starting
at degree zero and removing the error in each successive degree
proves the required jet surjection. The argument takes place in the
local ring and never divides by factorials.
-Put $d_p=(p-k_0N_p)q$, so $0le d_ple(k_0-1)q$. Among the sections
+Put d_p=(p-k_0N_p)q, so 0le d_ple(k_0-1)q. Among the sections
fixed before the reduction, choose
[
u_0in H^0(W,OO_W(T_0)),
- qquad
+
v_{d_p}in H^0(W,OO_W(T_0+d_pS))
]
-nonzero at the first and second base coordinates of $z_*$,
+nonzero at the first and second base coordinates of z_*,
respectively. Their tensor product belongs to the summand of
first-slot weight zero in the projective-bundle formula; its fibre
-monomial is the $d_p$-th power of the second coordinate. It defines
-a section of $d_pxi+T_{0,1}+T_{0,2}$ nonzero at $z_*$, because $z_*$
+monomial is the d_p-th power of the second coordinate. It defines
+a section of d_pxi+T_{0,1}+T_{0,2} nonzero at z_*, because z_*
lies off both axes. Multiplying the product jet system by this
-section changes its divisor from $N_pk_0M$ to
-$pqxi+B_{p,1}+B_{p,2}$. Multiplication is invertible on the local
+section changes its divisor from N_pk_0M to
+pqxi+B_{p,1}+B_{p,2}. Multiplication is invertible on the local
jet algebra. Only the finitely many previously fixed values of
-$d_p$ occur.
+d_p occur.
-Since $dim Z=2n+1$, the quotient by the $p$-th powers of regular
-parameters has top total degree $(2n+1)(p-1)$. For large $p$, this
-is at most $(2n+2)k_0N_p$. The underlying ordinary/Frobenius ideal
+Since dim Z=2n+1, the quotient by the p-th powers of regular
+parameters has top total degree (2n+1)(p-1). For large p, this
+is at most (2n+2)k_0N_p. The underlying ordinary/Frobenius ideal
comparison is also recorded in
cite[proof of Proposition~2.12, Equation~(2.8)]{MustataSchwede}.
@@ -2129,17 +2115,17 @@ cite[proof of Proposition~2.12, Equation~(2.8)]{MustataSchwede}.
## Full rank from the two-slot jets
-Let $Fcolon Wto W'$ be the relative Frobenius to the base-field
-twist. It is finite flat because $W$ is smooth. Write $S'$ for the
-twist of $S$, so $F^*S'=pS$, and define
+Let Fcolon Wto W' be the relative Frobenius to the base-field
+twist. It is finite flat because W is smooth. Write S' for the
+twist of S, so F^*S'=pS, and define
[
mathcal E=F_*calE=F_*OO_W(B_p),
- qquad s=rkmathcal E=p^n,calE=p^n,
- qquad R=s^2.
+ s=rkmathcal E=p^n,calE=p^n,
+ R=s^2.
]
-For integers $0le ale q$, put
-$U_a=H^0(W,OO_W(B_p+paS))$. The projection formula gives an
-evaluation map on $W'times W'$:
+For integers 0le ale q, put
+U_a=H^0(W,OO_W(B_p+paS)). The projection formula gives an
+evaluation map on W'times W':
⊕_{substack{a+b=qa,bge0}}
U_a ⊗ U_b ⊗
@@ -2148,41 +2134,41 @@ evaluation map on $W'times W'$:
-The map in (fr:two-slot-evaluation) has generic rank $R$.
+The map in (fr:two-slot-evaluation) has generic rank R.
Proof.
-Trivialize the two summands defining $Z$ near the selected base
-points. Let $z$ be the torus coordinate, whose value at $z_*$ is
-$z_0ne0$. By the projective-bundle formula, the sections in
+Trivialize the two summands defining Z near the selected base
+points. Let z be the torus coordinate, whose value at z_* is
+z_0ne0. By the projective-bundle formula, the sections in
Lemma~fr:positive-characteristic-jets are weight polynomials
-in $z$ with weight-$j$ coefficients in
+in z with weight-j coefficients in
[
H^0(W,B_p+jS) ⊗ H^0(W,B_p+(pq-j)S),
- qquad 0le jle pq.
+ 0le jle pq.
]
-Let $A_*$ be the tensor product of the local algebras of the two base
+Let A_* be the tensor product of the local algebras of the two base
Frobenius fibres. The local quotient in
Lemma~fr:positive-characteristic-jets is
[
A_*[z]/(z^p-z_0^p),
]
-free over $A_*$ with basis $1,z,...,z^{p-1}$. Project to the
-coefficient of $1$ in this basis. Exactly the weights $j=pa$
-survive, with multipliers $z_0^{pa}$. Their coefficients are the
+free over A_* with basis 1,z,...,z^{p-1}. Project to the
+coefficient of 1 in this basis. Exactly the weights j=pa
+survive, with multipliers z_0^{pa}. Their coefficients are the
values of the corresponding summands of
(fr:two-slot-evaluation), in the chosen frames and the
induced twisted frames. As the full jet map is surjective, these
-coefficients span $A_*$, of dimension $p^{2n}=R$. Thus the
+coefficients span A_*, of dimension p^{2n}=R. Thus the
evaluation has full rank at this pair of base points, and hence
generically. The calculation uses a basis over the possibly
-nonreduced algebra $A_*$, so reducedness of the Frobenius fibres is
+nonreduced algebra A_*, so reducedness of the Frobenius fibres is
unnecessary.
-The jet calculation at $z_*$ proves generic full rank; the diagonal
-estimate below will be tested at $(x',x')$, from the separately
-fixed point $x$.
+The jet calculation at z_* proves generic full rank; the diagonal
+estimate below will be tested at (x',x'), from the separately
+fixed point x.
## Sections on the Frobenius diagonal
@@ -2194,60 +2180,60 @@ The Frobenius fiber product has the cartesian square
W arrow[r,"F"'] & W'.
-Its reduced subscheme is the diagonal $W$. Set
+Its reduced subscheme is the diagonal W. Set
[
mathcal L_p=calL_p=OO_{Delta_F}(B_{p,1}+B_{p,2}+pqS_1).
]
-Write $h=Fcircpr_1=Fcircpr_2colon
-Delta_Fto W'$. On this fiber product,
+Write h=Fcircpr_1=Fcircpr_2colon
+Delta_Fto W'. On this fiber product,
[
OO(pS_1)simeq h^*OO_{W'}(S')simeqOO(pS_2).
]
-Consequently every pair of weights $a+b=q$ gives the same line bundle:
+Consequently every pair of weights a+b=q gives the same line bundle:
[
OO_{Delta_F}(B_{p,1}+B_{p,2}+paS_1+pbS_2)
simeqmathcal L_p.calL_p.
]
-On the diagonal of $W'times W'$, every source twist of
-(fr:two-slot-evaluation) becomes $-qS'$. Finite base change
+On the diagonal of W'times W', every source twist of
+(fr:two-slot-evaluation) becomes -qS'. Finite base change
and the projection formula identify the twisted target as
[
(mathcal EcalE ⊗ mathcal E)(qS')calE)(qS')simeq h_*mathcal L_p.calL_p.
]
The product sections defining the columns restrict to global
-sections of this one bundle $mathcal L_p$.calL_p$. Their values at a
+sections of this one bundle mathcal L_p.calL_p. Their values at a
diagonal point therefore lie in the image of
-$H^0(Delta_F,mathcal L_p)$calL_p)$ in the corresponding fibre of
-$h_*mathcal L_p$.calL_p$. This proves that the rank at every diagonal point
+H^0(Delta_F,mathcal L_p)calL_p) in the corresponding fibre of
+h_*mathcal L_p.calL_p. This proves that the rank at every diagonal point
is at most
h^0(Delta_F,mathcal L_p).calL_p).
-For a rank-$n$ bundle $V$ in characteristic $p$, denote by $A_j(V)$
-the degree-$j$ part of its symmetric algebra modulo the $p$-th powers
+For a rank-n bundle V in characteristic p, denote by A_j(V)
+the degree-j part of its symmetric algebra modulo the p-th powers
of local generators. This construction is independent of frame: the
-$p$-th power of a linear combination is the sum of the $p$-th powers
-in characteristic $p$. Each $A_j(V)$ is locally free and a quotient
-of $V^{ ⊗ j}$. Set
+p-th power of a linear combination is the sum of the p-th powers
+in characteristic p. Each A_j(V) is locally free and a quotient
+of V^{ ⊗ j}. Set
[
- u=n(p-1),qquad e_j=rk A_j(V),qquad
+ u=n(p-1), e_j=rk A_j(V),
sum_{j=0}^u e_j=p^n.
]
-Filtering $mathcal L_p$calL_p$ by powers of the ideal of the reduced
-diagonal in $Delta_F$ gives the graded bundles
+Filtering mathcal L_pcalL_p by powers of the ideal of the reduced
+diagonal in Delta_F gives the graded bundles
mathcal G_j=calG_j=OO_W(2B_p+pqS) ⊗ A_j(Ω_W^1),
- qquad 0le jle u.
+ 0le jle u.
Indeed, in smooth local coordinates, the algebra is generated by
-parameter differences with their $p$-th powers zero, and its
-degree-one conormal is $Ω_W^1$. This coordinate calculation can
+parameter differences with their p-th powers zero, and its
+degree-one conormal is Ω_W^1. This coordinate calculation can
be performed 'etale locally or in completions and identifies the
global associated graded with (fr:diagonal-graded). This is
the diagonal ideal form of the canonical Frobenius filtration
cite[Corollary~3.5]{KS}. The same calculation, using only the
-bundle from the second factor, filters $F^*mathcal E$calE$ with pieces
+bundle from the second factor, filters F^*mathcal EcalE with pieces
cite[Theorem~3.7]{Sun}
OO_W(B_p) ⊗ A_j(Ω_W^1).
@@ -2258,70 +2244,70 @@ Complementary multiplication is a perfect pairing:
A_{u-j}(V)simeq A_j(V)^vee ⊗ (det V)^{p-1}.
On monomials, each exponent vector is paired with its complement to
-$(p-1,...,p-1)$. The top line transforms by the character
-$(det V)^{p-1}$: this is immediate on diagonal matrices and hence
-identifies the character of $GL_n$ on that line.
+(p-1,...,p-1). The top line transforms by the character
+(det V)^{p-1}: this is immediate on diagonal matrices and hence
+identifies the character of GL_n on that line.
## A uniform bound for the diagonal rank
We bound the sections of every graded bundle in
(fr:diagonal-graded) by its rank times the same scalar
-polynomial in $p$. This uniformity lets us sum the ranks of the
+polynomial in p. This uniformity lets us sum the ranks of the
graded pieces without an additional factor for their number. We
begin with a slope bound on the fixed curve; here slope means
ordinary degree divided by rank.
-There is a constant $cge0$, independent of the sufficiently large
-residual characteristic $p$ and of $j$, such that for
-$V=Ω_W^1|_C$ one has
+There is a constant cge0, independent of the sufficiently large
+residual characteristic p and of j, such that for
+V=Ω_W^1|_C one has
[
mu_{max}(A_j(V))le(p-1)K_Wcdot C+nc.
]
Proof.
-Write $F_C$ for the absolute Frobenius of $C$. Langer's instability
+Write F_C for the absolute Frobenius of C. Langer's instability
estimate cite[Corollary~2.5]{Langer} gives
L_{min}(V):=lim_{eto∞}p^{-e}mu_{min}(F_C^{e*}V)
ge-(c)/(p-1).
-To make the uniformity explicit, let $g$ be the fixed genus of $C$.
-One can choose a nef line bundle $A_C$ of degree at most
-$d_g=max{4g-2,0}$ such that $T_C ⊗ A_C$ is globally
-generated; for $gle1$, take $A_C=OO_C$. Langer's bound is
+To make the uniformity explicit, let g be the fixed genus of C.
+One can choose a nef line bundle A_C of degree at most
+d_g=max{4g-2,0} such that T_C ⊗ A_C is globally
+generated; for gle1, take A_C=OO_C. Langer's bound is
[
mu_{min}(V)-L_{min}(V)
le((rk V-1)deg A_C)/(p-1).
]
-Since $rk V=n$ and $mu_{min}(V)ge0$, we may take
-$c=(n-1)d_g$, independently of the reduction.
+Since rk V=n and mu_{min}(V)ge0, we may take
+c=(n-1)d_g, independently of the reduction.
-We also need, for every integer $ige0$,
+We also need, for every integer ige0,
mu_{min}(V^{ ⊗ i})ge iL_{min}(V).
-Fix $p$ and $i$. Choose a line bundle $A_e$ of degree
-$-mu_{min}(F_C^{e*}V)+O(1)$, rounded up with a fixed
+Fix p and i. Choose a line bundle A_e of degree
+-mu_{min}(F_C^{e*}V)+O(1), rounded up with a fixed
genus-dependent additive constant, so that
-$F_C^{e*}V ⊗ A_e$ is globally generated. This follows from
+F_C^{e*}V ⊗ A_e is globally generated. This follows from
Serre duality: after subtracting any point, its minimum slope can
-be made greater than $2g(C)-2$, which annihilates $H^1$ and makes
+be made greater than 2g(C)-2, which annihilates H^1 and makes
evaluation at that point surjective. For every vector-bundle
-quotient $Q$ of $V^{ ⊗ i}$, the bundle
-$F_C^{e*}Q ⊗ A_e^{ ⊗ i}$ is then globally generated.
+quotient Q of V^{ ⊗ i}, the bundle
+F_C^{e*}Q ⊗ A_e^{ ⊗ i} is then globally generated.
Consequently
[
p^emu(Q)+ideg A_ege0.
]
-Divide by $p^e$ and let $eto∞$ to prove
-(fr:tensor-min). This limit is taken for each fixed $p,i$;
+Divide by p^e and let eto∞ to prove
+(fr:tensor-min). This limit is taken for each fixed p,i;
there is no interchange of Frobenius-iteration and characteristic
limits.
-Since $A_{u-j}(V)$ is a quotient of $V^{ ⊗ (u-j)}$, the perfect
+Since A_{u-j}(V) is a quotient of V^{ ⊗ (u-j)}, the perfect
pairing (fr:socle-pairing) gives
mu_{max}(A_j(V))
@@ -2331,47 +2317,47 @@ pairing (fr:socle-pairing) gives
-The divisors $B_p-pD=T_0-(p-k_0N_p)D$ range over a fixed finite
+The divisors B_p-pD=T_0-(p-k_0N_p)D range over a fixed finite
list. Combining (fr:intersection-bounds),
(fr:flag-degrees), and Lemma~fr:truncated-slope,
-therefore gives, uniformly for $0le jle u$,
+therefore gives, uniformly for 0le jle u,
mu_{max}(mathcal G_j|_C)calG_j|_C)
&le p(4+2/r)Λ H^n+O(1)notag
&le M_p:=7pΛ H^n+c_0,
-where $c_0ge0$ is a fixed constant. The coefficient $7$ provides
+where c_0ge0 is a fixed constant. The coefficient 7 provides
harmless room in this common bound.
-For all sufficiently large $p$,
+For all sufficiently large p,
[
h^0(Delta_F,mathcal L_p)calL_p)
le R(7^nH^n+O(1/p))<R/4.
]
-The error constant is independent of the filtration index $j$.
+The error constant is independent of the filtration index j.
Proof.
Write the fixed flag as
[
W=W_nsupset W_{n-1}supset...supset W_1=C,
- qquad W_{n-k}in|h_k|_{W_{n-k+1}}
- quad(1le kle n-1).
+ W_{n-k}in|h_k|_{W_{n-k+1}}
+ (1le kle n-1).
]
-At step $k$, fix the preceding nonnegative integers
-$b_1,...,b_{k-1}$ and put
+At step k, fix the preceding nonnegative integers
+b_1,...,b_{k-1} and put
[
mathcal F_k=calF_k=mathcal G_j|_calG_j|_{W_{n-k+1}}(-sum_{i<k}b_ih_i).
]
-For each $b_kge0$, the divisor restriction sequence is
+For each b_kge0, the divisor restriction sequence is
[
0→mathcal F_k(-(b_k+1)h_k)calF_k(-(b_k+1)h_k)
→mathcal F_k(-b_kh_k)calF_k(-b_kh_k)
→mathcal F_k(-b_kh_k)|_calF_k(-b_kh_k)|_{W_{n-k}}
→0.
]
-Iterating it for $b_k=0,1,...$ and then proceeding to the next
+Iterating it for b_k=0,1,... and then proceeding to the next
member of the flag bounds the section space by
h^0(W,mathcal G_j)calG_j)le
@@ -2379,14 +2365,14 @@ member of the flag bounds the section space by
h^0(C,mathcal G_j|_C(-calG_j|_C(-sum_i b_ih_i|_C)).
At each stage, the remainder is zero after sufficiently negative
-ample twisting. Its stopping index need not be uniform in $j,p$ or
+ample twisting. Its stopping index need not be uniform in j,p or
in the preceding twists: enlarging the nonnegative finite sums to
the displayed lattice sum preserves the inequality.
A summand in (fr:flag-lattice-sum) vanishes when
-$sum_i b_i(h_icdot C)>M_p$, because then its maximum slope is
-negative. Each nonzero summand has at most $e_j(M_p+1)$ sections.
-Indeed, evaluation at $lfloor M_prfloor+1$ distinct points is
+sum_i b_i(h_icdot C)>M_p, because then its maximum slope is
+negative. Each nonzero summand has at most e_j(M_p+1) sections.
+Indeed, evaluation at lfloor M_prfloor+1 distinct points is
injective; the kernel has negative maximum slope after subtracting
those points. By (fr:flag-degrees) it follows that
@@ -2395,41 +2381,41 @@ those points. By (fr:flag-degrees) it follows that
prod_{i=1}^{n-1}(1+(M_p)/(l_iΛ H^n))notag
&=e_jp^n(7^nH^n+O(1/p)).
-The leading coefficient follows from $Λ=prod_il_i$. More
+The leading coefficient follows from Λ=prod_il_i. More
importantly, the whole scalar polynomial on the first line is the
-same for every $j$.
+same for every j.
Sum over the filtration in (fr:diagonal-graded). The ranks
-satisfy $sum_j e_j=p^n$, so (fr:common-flag-polynomial) gives
-the asserted bound with $R=p^{2n}$. There is no extra factor for
+satisfy sum_j e_j=p^n, so (fr:common-flag-polynomial) gives
+the asserted bound with R=p^{2n}. There is no extra factor for
the number of graded pieces. Finally,
[
7^nH^nle(14ε)^n<1/4
]
by (fr:intersection-bounds), proving the strict inequality for
-all large $p$.
+all large p.
## The two orders of the determinant
-Choose $R$ generically independent columns of
+Choose R generically independent columns of
(fr:two-slot-evaluation) from bases of its source vector spaces.
Their determinant is a nonzero section
[
0nesigmain H^0(W'times W',mathcal Q_1calQ_1boxtimesmathcal Q_2)calQ_2)
]
of an actual external-product line bundle. To specify its factors,
-let $m_i$ be the sum of the weights of the $R$ chosen columns in slot
-$i$. The target determinant is
-$(detmathcal E)^scalE)^sboxtimes(detmathcal E)^s$,calE)^s$, and the chosen source
-determinant is $OO(-m_1S')boxtimesOO(-m_2S')$. Thus
+let m_i be the sum of the weights of the R chosen columns in slot
+i. The target determinant is
+(detmathcal E)^scalE)^sboxtimes(detmathcal E)^s,calE)^s, and the chosen source
+determinant is OO(-m_1S')boxtimesOO(-m_2S'). Thus
[
mathcal Q_i=(calQ_i=(detmathcal E)^calE)^{ ⊗ s}
⊗ OO_{W'}(m_iS'),
- qquad lambda_i=m_i/R.
+ lambda_i=m_i/R.
]
-Every column weight lies between $0$ and $q$, so $0lelambda_ile q$,
+Every column weight lies between 0 and q, so 0lelambda_ile q,
independently of the columns chosen.
Under the numerical identification with the base-field twist,
@@ -2437,51 +2423,51 @@ Under the numerical identification with the base-field twist,
frac{c_1(mathcal Q_i)}calQ_i)R
=frac{c_1(mathcal E)}calE)s+lambda_iS
=frac{B_p}p+(p-1)/(2p)K_W+lambda_iS,
- qquad 0lelambda_ile q.
+ 0lelambda_ile q.
The Frobenius first-Chern-class identity is also
cite[Lemma~4.2]{Sun}; it is an identity of rational divisor classes.
For the second equality, use (fr:Frob-pullback-graded).
Complementary degrees in (fr:socle-pairing) have total first
-Chern class $e_j(p-1)K_W$. Summing gives
+Chern class e_j(p-1)K_W. Summing gives
[
sum_jc_1(A_j(Ω_W^1))=frac{s(p-1)}2K_W.
]
-Thus $c_1(F^*mathcal E)=sB_p+s(p-1)K_W/2$,calE)=sB_p+s(p-1)K_W/2$, and Frobenius pullback of
-twisted divisor classes multiplies by $p$, as required.
+Thus c_1(F^*mathcal E)=sB_p+s(p-1)K_W/2,calE)=sB_p+s(p-1)K_W/2, and Frobenius pullback of
+twisted divisor classes multiplies by p, as required.
-For $0lelambdale q$, the fixed movable inequality (fr:movable-input)
+For 0lelambdale q, the fixed movable inequality (fr:movable-input)
gives
[
(D+K_W/2+lambda S)cdotgammale40ε Jcdotgamma,
]
-where pullback by $pi_x$ is understood. This concerns only
+where pullback by pi_x is understood. This concerns only
intersections of fixed divisors and the fixed complete-intersection
witness, so it remains true on the reduction. The class in
(fr:slot-determinant-class) differs from its left-hand divisor
-at $lambda=lambda_i$ by
+at lambda=lambda_i by
[
frac1p(B_p-pD-K_W/2).
]
Its numerator belongs to the fixed finite list
-$T_0-aD-K_W/2$, $ain{0,...,k_0-1}$. The resulting error after
-intersection with $gamma$ is therefore $O(p^{-1})$, uniformly over
+T_0-aD-K_W/2, ain{0,...,k_0-1}. The resulting error after
+intersection with gamma is therefore O(p^{-1}), uniformly over
the selected determinant columns.
-Let $x'$ be the twisted point, and let $0netau_iin H^0(W',mathcal Q_i)$.calQ_i)$.
-The strict transform of the effective divisor of $tau_i$ on the
-blowup at $x'$ pairs nonnegatively with the twist of $gamma$. Since
-$Jcdotgamma>0$, it follows that
+Let x' be the twisted point, and let 0netau_iin H^0(W',mathcal Q_i).calQ_i).
+The strict transform of the effective divisor of tau_i on the
+blowup at x' pairs nonnegatively with the twist of gamma. Since
+Jcdotgamma>0, it follows that
frac{ord_{x'}(tau_i)}Rle40ε+O(1/p).
-The constant is uniform in the columns and in $tau_i$. This
+The constant is uniform in the columns and in tau_i. This
argument tests sections on the reduction itself; it does not require
them to lift to characteristic zero.
-K"unneth identifies the space containing $sigma$ with
-$H^0(W',mathcal Q_1)calQ_1) ⊗ H^0(W',mathcal Q_2)$.calQ_2)$. Choose bases
-adapted to the orders of vanishing at $x'$ in each factor. In each
+K"unneth identifies the space containing sigma with
+H^0(W',mathcal Q_1)calQ_1) ⊗ H^0(W',mathcal Q_2).calQ_2). Choose bases
+adapted to the orders of vanishing at x' in each factor. In each
degree, leading terms are linearly independent. Tensor products of
these leading terms remain independent in each bidegree, since the
two sets of parameters are disjoint; terms of different bidegrees
@@ -2493,83 +2479,83 @@ the sum of the two maximum basis orders. Applying
On the other hand, (fr:diagonal-rank-upper) and
-Lemma~fr:diagonal-sections bound the matrix rank at $(x',x')$
-by a number strictly smaller than $R/4$. Trivialize its line bundles
+Lemma~fr:diagonal-sections bound the matrix rank at (x',x')
+by a number strictly smaller than R/4. Trivialize its line bundles
near that point. Invertible constant row operations make more than
-$3R/4$ rows vanish at the point. Each entry of those rows belongs to
+3R/4 rows vanish at the point. Each entry of those rows belongs to
the maximal ideal, so every term of the determinant has order at
-least $3R/4$. Therefore
+least 3R/4. Therefore
ord_{(x',x')}sigmage3R/4.
Equations (fr:determinant-order-upper) and
-(fr:determinant-order-lower) contradict $80ε<3/4$ for
-sufficiently large $p$.
+(fr:determinant-order-lower) contradict 80ε<3/4 for
+sufficiently large p.
This contradiction proves Theorem~thm:finite-data. The
-full-rank calculation used $z_*$, while the diagonal estimate holds
-at every point and was tested at the separately chosen $x$. The
+full-rank calculation used z_*, while the diagonal estimate holds
+at every point and was tested at the separately chosen x. The
errors were controlled by one fixed finite list of divisor classes
and one fixed flag polynomial. Thus their constants do not depend
-on $p$, the filtration index, or the chosen columns.
+on p, the filtration index, or the chosen columns.
## Application to the terminal case
-We return to Theorem~thm:terminal-big. Thus $X$ is a projective
-terminal $Q$-factorial variety of dimension $nge2$ with $K_Xsim0$,
-and $L$ is a nef Cartier divisor of full nef dimension. Assume $L$
-is not big. Recall the ample rational divisors $P=r(L+tA)$, where
-$A$ is a fixed ample Cartier divisor and $t>0$ tends to zero, with
+We return to Theorem~thm:terminal-big. Thus X is a projective
+terminal Q-factorial variety of dimension nge2 with K_Xsim0,
+and L is a nef Cartier divisor of full nef dimension. Assume L
+is not big. Recall the ample rational divisors P=r(L+tA), where
+A is a fixed ample Cartier divisor and t>0 tends to zero, with
[
r→∞,
- qquad P^n→ε^n,
- qquad rho=(P^n)^{1/n}.
+ P^n→ε^n,
+ rho=(P^n)^{1/n}.
]
-The positive rational number $ε$ was chosen so that
-$(14ε)^n<1/4$ and $80ε<3/4$. Choose the integer $q$
+The positive rational number ε was chosen so that
+(14ε)^n<1/4 and 80ε<3/4. Choose the integer q
supplied by Proposition~prop:large-seshadri, and then choose
-$t$ sufficiently small that this proposition and
-Lemma~lemma:scalar-bound apply, that $r>q+1$, and that
-$P^n<(2ε)^n$. We now fix $t$, and therefore also $P,r,rho$
+t sufficiently small that this proposition and
+Lemma~lemma:scalar-bound apply, that r>q+1, and that
+P^n<(2ε)^n. We now fix t, and therefore also P,r,rho
and the projective bundle in (eq:bundle). Every remaining
-choice is made over $C$ before reducing to positive characteristic.
+choice is made over C before reducing to positive characteristic.
The divisors and the flag.: Choose a smooth projective
-resolution $wcolon Wto X$ and set
+resolution wcolon Wto X and set
[
- D=w^*P,qquad S=w^*L,qquad H_0=w^*P.
+ D=w^*P, S=w^*L, H_0=w^*P.
]
-The divisor $S$ is integral Cartier, and $H_0$ is nef and big. Since
-$X$ is terminal and $K_Xsim0$, there is an integral effective
-$w$-exceptional divisor $E$ with $K_Wsim E$. The projection formula
-and the nefness of $P-rL$ give
+The divisor S is integral Cartier, and H_0 is nef and big. Since
+X is terminal and K_Xsim0, there is an integral effective
+w-exceptional divisor E with K_Wsim E. The projection formula
+and the nefness of P-rL give
[
K_WH_0^{n-1}=0,
- qquad
+
(2D+qS)H_0^{n-1}le(2+q/r)H_0^n.
]
-At $H_0$, the inequalities
-$K_WH_0^{n-1}<2H_0^n/r$ and $(2D+qS)H_0^{n-1}<4H_0^n$ are strict.
+At H_0, the inequalities
+K_WH_0^{n-1}<2H_0^n/r and (2D+qS)H_0^{n-1}<4H_0^n are strict.
Thus, by continuity of intersection products, a sufficiently small
-rational ample perturbation $H=H_0+eta A_W$, with $A_W$ an ample
-Cartier divisor on $W$ and $eta>0$, satisfies
+rational ample perturbation H=H_0+eta A_W, with A_W an ample
+Cartier divisor on W and eta>0, satisfies
[
H^nle(2ε)^n,
- qquad
+
K_WH^{n-1}le 2H^n/r,
- qquad
+
(2D+qS)H^{n-1}le4H^n.
]
This gives (fr:intersection-bounds).
-The canonical divisor of $W$ is pseudo-effective. Generic
+The canonical divisor of W is pseudo-effective. Generic
semipositivity of the cotangent bundle, originating in [cite]
and used here in the form cite[Theorem~2.1]{CP}, together with
restriction to sufficiently general complete-intersection curves
cite[Theorem~6.1]{MR}, gives a smooth integral flag with successive
-divisor classes $l_iH$, for sufficiently large divisible integers
-$l_i$, such that
+divisor classes l_iH, for sufficiently large divisible integers
+l_i, such that
[
mu_{min}(Ω_W^1|_C)ge0.
]
@@ -2577,55 +2563,55 @@ This is the flag in Hypothesis~(i) of
Theorem~thm:finite-data.
The finite jet system.: On the bundle in (eq:bundle),
-write $M=D_1+D_2+qxi$. The divisor $M$ is ample: $xi$ is nef and
-relatively ample, and $D_1+D_2$ is the pullback of an ample divisor on
-$Xtimes X$. Proposition~prop:large-seshadri supplies a smooth
-point $z_*$ in the complement of the two axes with
+write M=D_1+D_2+qxi. The divisor M is ample: xi is nef and
+relatively ample, and D_1+D_2 is the pullback of an ample divisor on
+Xtimes X. Proposition~prop:large-seshadri supplies a smooth
+point z_* in the complement of the two axes with
[
ε(M;z_*)>2n+2.
]
-The point can be chosen over the open set where $w$ is an
-isomorphism. Choose a rational number $a$ strictly between
-$2n+2$ and $ε(M;z_*)$. On the blowup
-$widetilde Zto Z$ at $z_*$, with exceptional divisor $G$, the
-rational divisor $b^*M-aG$ is ample. For sufficiently large
-divisible $k_0$, Serre vanishing gives
+The point can be chosen over the open set where w is an
+isomorphism. Choose a rational number a strictly between
+2n+2 and ε(M;z_*). On the blowup
+widetilde Zto Z at z_*, with exceptional divisor G, the
+rational divisor b^*M-aG is ample. For sufficiently large
+divisible k_0, Serre vanishing gives
[
H^1(widetilde Z,OO_{widetilde Z}(k_0b^*M-k_0aG))=0.
]
-Since $z_*$ is smooth, the sections on the thickened exceptional
-divisor $k_0aG$ identify with the jet quotient at $z_*$ modulo
-$mathfrak m_frakm_{z_*}^{k_0a}$, in a local frame for $k_0M$. The
-restriction sequence therefore shows that sections of $k_0M$
-generate jets through order $k_0a-1$, and hence through order
-$(2n+2)k_0$. Increase the divisibility of $k_0$ to make $k_0P$
+Since z_* is smooth, the sections on the thickened exceptional
+divisor k_0aG identify with the jet quotient at z_* modulo
+mathfrak m_frakm_{z_*}^{k_0a}, in a local frame for k_0M. The
+restriction sequence therefore shows that sections of k_0M
+generate jets through order k_0a-1, and hence through order
+(2n+2)k_0. Increase the divisibility of k_0 to make k_0P
Cartier. Pulling these sections to
[
Z_W=PP(OO_W(S)_1 ⊕ OO_W(S)_2)
→ Wtimes W
]
preserves the jets at the corresponding point, because the induced
-map $Z_Wto Z$ is an isomorphism near that point. The pulled-back
-divisor is $D_1+D_2+qxi$. We have obtained the finite system
+map Z_Wto Z is an isomorphism near that point. The pulled-back
+divisor is D_1+D_2+qxi. We have obtained the finite system
required in Hypothesis~(ii).
The movable curve class.: Choose a very general point
-$xin W$ outside the exceptional locus, with image in the locus where
+xin W outside the exceptional locus, with image in the locus where
Lemma~lemma:scalar-bound holds. Let
-$pi_xcolonwidehatwidehat WWto W$ be the blowup at $x$ and $J$ its
+pi_xcolonwidehatwidehat WWto W be the blowup at x and J its
exceptional divisor. The rational divisor
[
D^+=2D+E
]
-is big. For every sufficiently divisible positive integer $m$,
-normality of $X$ and exceptionality of $E$ give
-$w_*OO_W(mE)=OO_X$. Thus every section of $mD^+$ comes from a
-section of $2mP$, multiplied by the section of the fixed divisor
-$mE$. Since $xnotinSupp E$, Lemma~lemma:scalar-bound
+is big. For every sufficiently divisible positive integer m,
+normality of X and exceptionality of E give
+w_*OO_W(mE)=OO_X. Thus every section of mD^+ comes from a
+section of 2mP, multiplied by the section of the fixed divisor
+mE. Since xnotinSupp E, Lemma~lemma:scalar-bound
implies
ord_x(s)/mle8rho<16ε
- qquad(0ne sin H^0(W,OO_W(mD^+))).
+ (0ne sin H^0(W,OO_W(mD^+))).
It follows that
@@ -2633,8 +2619,8 @@ It follows that
pi_x^*D^+-40ε J is not pseudo-effective.
Indeed, if it were pseudo-effective, a convex combination with the
-big divisor $pi_x^*D^+$ would make $pi_x^*D^+-cJ$ big for any
-rational $c$ with $16ε<c<40ε$. A nonzero section of a
+big divisor pi_x^*D^+ would make pi_x^*D^+-cJ big for any
+rational c with 16ε<c<40ε. A nonzero section of a
sufficiently divisible multiple would then give a section violating
(fr:application-order).
@@ -2643,9 +2629,9 @@ movable curves [cite], the strict separation
in~(fr:application-not-pe) is detected by a class
[
gamma=f_*(A_1... A_{n-1}),
- qquad fcolon V→widehatwidehat W,W,
+ fcolon V→widehatwidehat W,W,
]
-where $V$ is smooth projective, $f$ is birational, and the $A_i$ are
+where V is smooth projective, f is birational, and the A_i are
ample integral Cartier divisors. To obtain this form, first choose a
generating strongly movable class with negative pairing, then
approximate its ample real classes by rational ones and clear
@@ -2653,16 +2639,16 @@ denominators. We have
[
(pi_x^*D^+)cdotgamma<40ε Jcdotgamma.
]
-The left side is nonnegative, because $pi_x^*D^+$ is
-pseudo-effective, so $Jcdotgamma>0$.
+The left side is nonnegative, because pi_x^*D^+ is
+pseudo-effective, so Jcdotgamma>0.
-For every $0lelambdale q$,
+For every 0lelambdale q,
[
D^+-(D+K_W/2+lambda S)sim_R w^*(P-lambda L)+E/2
]
-is pseudo-effective: $P-lambda L=(P-rL)+(r-lambda)L$ is nef, and
-$E$ is effective. Pulling back to $widehatwidehat W$W$ and pairing with
-$gamma$ therefore gives
+is pseudo-effective: P-lambda L=(P-rL)+(r-lambda)L is nef, and
+E is effective. Pulling back to widehatwidehat WW and pairing with
+gamma therefore gives
[
pi_x^*(D+K_W/2+lambda S)cdotgamma
le(pi_x^*D^+)cdotgamma
@@ -2670,32 +2656,32 @@ $gamma$ therefore gives
]
This proves Hypothesis~(iii). All hypotheses of
Theorem~thm:finite-data now hold for one fixed collection of
-complex data, a contradiction. Hence $L$ is big, completing the
+complex data, a contradiction. Hence L is big, completing the
proof of Theorem~thm:terminal-big.
# Completion of the full-dimension step
By Proposition~prop:reduction-cy, Theorem~thm:key-ingredient
-reduces to the following claim in dimension $n$:
+reduces to the following claim in dimension n:
- (X,B) projective klt,qquad K_X+Bequiv0,
- M nef and $Q$-Cartier of full nef dimension
- quad⇒quad M big.
+ (X,B) projective klt, K_X+Bequiv0,
+ M nef and Q-Cartier of full nef dimension
+ ⇒ M big.
-Theorem~thm:terminal-big proves this claim for terminal $X$
-with $B=0$. We now extend it in two steps: first to arbitrary
-klt $X$ with $B=0$, using an index-one cover and a crepant
-terminalization; then to arbitrary effective $B$, using a small
+Theorem~thm:terminal-big proves this claim for terminal X
+with B=0. We now extend it in two steps: first to arbitrary
+klt X with B=0, using an index-one cover and a crepant
+terminalization; then to arbitrary effective B, using a small
perturbation of the boundary and Lemma~lemma:two-nef-model.
## The case of zero boundary
-Suppose $X$ is klt with $K_Xequiv 0$. By
-Theorem~input:abundance, $K_X$ is semiample, hence
-$K_Xsim_Q0$. Indeed, a semiample numerically trivial divisor
+Suppose X is klt with K_Xequiv 0. By
+Theorem~input:abundance, K_X is semiample, hence
+K_Xsim_Q0. Indeed, a semiample numerically trivial divisor
defines a morphism with zero-dimensional image, so a sufficiently
divisible multiple is linearly trivial.
@@ -2704,7 +2690,7 @@ Take the global index-one cover
picolon X^{ind}→ X
]
associated with a trivialization of the smallest positive multiple
-of $K_X$ that is linearly trivial. This is a finite quasi-'etale
+of K_X that is linearly trivial. This is a finite quasi-'etale
cover with
[
K_{X^{ind}}=pi^*K_Xsim0.
@@ -2717,51 +2703,51 @@ morphism
[
hcolon widetilde X→ X^{ind}
]
-with $widetilde X$ terminal and $Q$-factorial and
-$K_{widetilde X}=h^*K_{X^{ind}}sim 0$.
+with widetilde X terminal and Q-factorial and
+K_{widetilde X}=h^*K_{X^{ind}}sim 0.
-Set $widetilde M=(picirc h)^*M$. This divisor is nef. To verify
+Set widetilde M=(picirc h)^*M. This divisor is nef. To verify
that it has full nef dimension, choose a very general point upstairs
-outside the exceptional locus of $h$ and above the very general
-locus used for $M$. Every curve through this point maps to a curve
-through a point where $M$ has positive degree, so $widetilde M$ has
+outside the exceptional locus of h and above the very general
+locus used for M. Every curve through this point maps to a curve
+through a point where M has positive degree, so widetilde M has
positive degree there as well by the projection formula. A positive
-Cartier multiple of $widetilde M$ is therefore big by
+Cartier multiple of widetilde M is therefore big by
Theorem~thm:terminal-big. Bigness is invariant under a
-dominant generically finite pullback, so $M$ is big on $X$.
+dominant generically finite pullback, so M is big on X.
## Introducing the boundary
-Now let $(X,B)$ satisfy (eq:completion-cy-claim) with $Bne0$.
+Now let (X,B) satisfy (eq:completion-cy-claim) with Bne0.
By Theorem~input:abundance,
K_X+Bsim_Q0.
-Choose a rational $alpha>0$ sufficiently small that
-$(X,(1+alpha)B)$ is klt, and set
+Choose a rational alpha>0 sufficiently small that
+(X,(1+alpha)B) is klt, and set
[
T_*=K_X+(1+alpha)Bsim_Qalpha B.
]
-Then $T_*$ is pseudo-effective and $kappa(T_*)ge0$. The full
-condition holds for $T_*$ and $M$: at a curve $C$ through a point
-outside $Supp B$, $T_*cdot C=alpha Bcdot Cge0$, while
-$Mcdot C>0$ if that point is very general.
+Then T_* is pseudo-effective and kappa(T_*)ge0. The full
+condition holds for T_* and M: at a curve C through a point
+outside Supp B, T_*cdot C=alpha Bcdot Cge0, while
+Mcdot C>0 if that point is very general.
-We do not use (P$_n$) here; the argument uses only
-Lemma~lemma:two-nef-model and the $B=0$ case just proved.
+We do not use (P_n) here; the argument uses only
+Lemma~lemma:two-nef-model and the B=0 case just proved.
-If $kappa(K_X+B)>0$, Proposition~prop:full-positive applied
-directly to the pair $(X,B)$ with $M$ gives $K_X+B+cM$ big for every
-$c>0$; since $K_X+Bequiv0$, this makes $M$ big. We may therefore
-assume $kappa(K_X+B)=0$, so that $K_X+Bsim_Q0$. Then $kappa(T_*)$
+If kappa(K_X+B)>0, Proposition~prop:full-positive applied
+directly to the pair (X,B) with M gives K_X+B+cM big for every
+c>0; since K_X+Bequiv0, this makes M big. We may therefore
+assume kappa(K_X+B)=0, so that K_X+Bsim_Q0. Then kappa(T_*)
may still be positive or zero; we treat the two cases with the same
machinery below.
Apply
-Lemma~lemma:two-nef-model to $(X,(1+alpha)B)$ and $M$. Write
-$Y$ for its final model, and let $B_Y$, $N$ denote the transforms of
-$B$ and $M$. Set $U=K_Y+(1+alpha)B_Y$. The lemma gives
-$Usim_Q0$ and, for some rational $s>0$, that $U+sN$ is nef of full
+Lemma~lemma:two-nef-model to (X,(1+alpha)B) and M. Write
+Y for its final model, and let B_Y, N denote the transforms of
+B and M. Set U=K_Y+(1+alpha)B_Y. The lemma gives
+Usim_Q0 and, for some rational s>0, that U+sN is nef of full
nef dimension. The maps in the construction extract no divisors, so
pushing forward the rational linear equivalence (eq:cy-cy)
also gives
@@ -2769,96 +2755,95 @@ also gives
K_Y+B_Ysim_Q0.
]
Subtracting these two equivalences shows that
-$alpha B_Ysim_Q0$. Since $B_Y$ is effective on a projective
+alpha B_Ysim_Q0. Since B_Y is effective on a projective
variety, intersection with a sufficiently ample complete-intersection
-curve forces $B_Y=0$. The pair $(Y,0)$ is therefore klt with
-$K_Ysim_Q0$, and $N$ is nef of full nef dimension (because
-$U+sNsim_Q sN$). The zero-boundary case just proved makes $N$
+curve forces B_Y=0. The pair (Y,0) is therefore klt with
+K_Ysim_Q0, and N is nef of full nef dimension (because
+U+sNsim_Q sN). The zero-boundary case just proved makes N
big.
-Consider now the fixed birational model $W$ with maps $p$ to $X$
-and $q$ to $Y$ from Lemma~lemma:two-nef-model. The effective
+Consider now the fixed birational model W with maps p to X
+and q to Y from Lemma~lemma:two-nef-model. The effective
exceptional comparisons are
[
p^*T_*=q^*U+E_0,
- qquad
+
p^*(T_*+sM)=q^*(U+sN)+E_1,
]
-with $E_0,E_1ge0$ and $q$-exceptional. Since $Usim_Q0$, we get
+with E_0,E_1ge0 and q-exceptional. Since Usim_Q0, we get
[
p^*(sM)=q^*(sN)+(E_1-E_0).
]
-The difference $D:=E_1-E_0=p^*(sM)-q^*(sN)$ is $q$-exceptional:
-its pushforward under $q$ is $sN-sN=0$. We claim $D$ is also
-$q$-nef. Indeed, if $C$ is any curve contracted by $q$, then
+The difference D:=E_1-E_0=p^*(sM)-q^*(sN) is q-exceptional:
+its pushforward under q is sN-sN=0. We claim D is also
+q-nef. Indeed, if C is any curve contracted by q, then
[
Dcdot C=p^*(sM)cdot C-q^*(sN)cdot C
=sMcdot p_*C-0ge0,
]
-because $M$ is nef and $q^*N$ is numerically trivial on the
-$q$-contracted $C$. The negativity lemma then gives $Dge0$.
+because M is nef and q^*N is numerically trivial on the
+q-contracted C. The negativity lemma then gives Dge0.
Hence
[
p^*(sM)ge q^*(sN)
]
-as actual divisors. If $N$ is big, so is $q^*N$, and the
-inequality makes $p^*(sM)$ big, hence $sM$ (and $M$) big.
+as actual divisors. If N is big, so is q^*N, and the
+inequality makes p^*(sM) big, hence sM (and M) big.
-In the case $kappa(T_*)>0$, Proposition~prop:full-positive
-applied to the pair $(X,(1+alpha)B)$ with $M$ gives $T_*+cM$ big
-for every $c>0$. Combined with the effective exceptional comparison
-of $p$ to $q$ from Lemma~lemma:two-nef-model, this yields
-bigness of $M$ on $X$.
+In the case kappa(T_*)>0, Proposition~prop:full-positive
+applied to the pair (X,(1+alpha)B) with M gives T_*+cM big
+for every c>0. Combined with the effective exceptional comparison
+of p to q from Lemma~lemma:two-nef-model, this yields
+bigness of M on X.
-In either case, then, $M$ is big. This proves
+In either case, then, M is big. This proves
(eq:completion-cy-claim), and hence Theorem~thm:key-ingredient
in all dimensions.
# Positive parts and numerical semiampleness
-We now prove (Z$_n$) by induction on $n$, using (P$_j$) and (Z$_j$)
-for $j<n$ as inductive hypotheses. The argument runs in four
+We now prove (Z_n) by induction on n, using (P_j) and (Z_j)
+for j<n as inductive hypotheses. The argument runs in four
steps. First, a lemma recording the standard rules for the
-positive part $P_sigma$. Second, a birational replacement by a
-model on which the adjoint $T+mL$ is nef for a suitable multiple
-$m$. Third, on that model either $T+mL$ has full nef dimension
-(so (P$_n$) applies and the good minimal model gives a semiample
+positive part P_sigma. Second, a birational replacement by a
+model on which the adjoint T+mL is nef for a suitable multiple
+m. Third, on that model either T+mL has full nef dimension
+(so (P_n) applies and the good minimal model gives a semiample
positive part) or its nef reduction is nontrivial (so we descend
to a lower-dimensional base, on which the induction applies).
Fourth, we descend the semiampleness from a smooth birational model
-to $X$ itself, using that $X$ has rational singularities.
+to X itself, using that X has rational singularities.
## Positive-part operations
-The positive part $P_sigma$ satisfies the following properties.
-
-item If $pcolon Vto X$ is a projective birational morphism from a
-smooth projective $V$ to a normal projective $X$, and $D$ is a
-pseudo-effective rational divisor on $X$, and $Ege0$ is
-$p$-exceptional, then
+The positive part P_sigma satisfies the following properties.
+ If pcolon Vto X is a projective birational morphism from a
+smooth projective V to a normal projective X, and D is a
+pseudo-effective rational divisor on X, and Ege0 is
+p-exceptional, then
[
P_sigma(p^*D+E)=P_sigma(p^*D).
]
-item If $fcolon Vto S$ is a surjective morphism of smooth
-projective varieties, $D$ is pseudo-effective on $S$, and
-$P_sigma(D)$ is nef, then
+ If fcolon Vto S is a surjective morphism of smooth
+projective varieties, D is pseudo-effective on S, and
+P_sigma(D) is nef, then
[
P_sigma(f^*D)=f^*P_sigma(D).
]
-item On a fixed smooth projective variety, rationality and
-numerical semiampleness of $P_sigma(D)$ depend only on the
-numerical class of $D$.
+ On a fixed smooth projective variety, rationality and
+numerical semiampleness of P_sigma(D) depend only on the
+numerical class of D.
Proof.
Parts (i) and (ii) are cite[Lemmas 2.4 and 2.5]{LP}, with the
target allowed to be singular in (i). For (iii): each asymptotic
-order $sigma_Gamma(D)$ depends only on the numerical class, so
+order sigma_Gamma(D) depends only on the numerical class, so
numerically equivalent divisors have the same negative part as an
-effective real divisor. For rational $Dequiv D'$,
+effective real divisor. For rational Dequiv D',
[
P_sigma(D')-P_sigma(D)=D'-D.
]
@@ -2874,224 +2859,223 @@ rational singularities. Recall that klt varieties have rational
singularities.
-Let $pcolon Wto X$ be a projective birational morphism from a
-smooth projective $W$ to a normal projective $X$ with rational
-singularities. Let $D$ be a $Q$-Cartier divisor on $X$. If
-$p^*D$ is numerically equivalent to a semiample $Q$-divisor on
-$W$, then $D$ is numerically equivalent to a semiample $Q$-divisor
-on $X$.
+Let pcolon Wto X be a projective birational morphism from a
+smooth projective W to a normal projective X with rational
+singularities. Let D be a Q-Cartier divisor on X. If
+p^*D is numerically equivalent to a semiample Q-divisor on
+W, then D is numerically equivalent to a semiample Q-divisor
+on X.
Proof.
-Let $A_W$ be a semiample rational divisor on $W$ with
-$A_Wequiv p^*D$. After clearing denominators, the difference of
+Let A_W be a semiample rational divisor on W with
+A_Wequiv p^*D. After clearing denominators, the difference of
the corresponding line bundles is numerically trivial. Numerically
-trivial line bundles modulo $Pic^0(W)$ are torsion. Thus, after
-raising a positive integer $r$, there is
+trivial line bundles modulo Pic^0(W) are torsion. Thus, after
+raising a positive integer r, there is
[
OO_W(rA_W)simeq p^*OO_X(rD) ⊗ P_W,
- qquad P_WinPic^0(W).
+ P_WinPic^0(W).
]
-Rational singularities identify $Pic^0(X)$ with $Pic^0(W)$ by
-pullback. Write $P_W=p^*P_X$ for $P_XinPic^0(X)$, and set
-$mathcal A_X=calA_X=OO_X(rD) ⊗ P_X$. Then
+Rational singularities identify Pic^0(X) with Pic^0(W) by
+pullback. Write P_W=p^*P_X for P_XinPic^0(X), and set
+mathcal A_X=calA_X=OO_X(rD) ⊗ P_X. Then
[
p^*mathcal A_XcalA_XsimeqOO_W(rA_W).
]
A positive power of the line bundle on the right is globally
-generated, because $A_W$ is semiample. Since $p_*OO_W=OO_X$ for
+generated, because A_W is semiample. Since p_*OO_W=OO_X for
a birational morphism from a smooth variety, the projection formula
-identifies the sections with those of the same power of $mathcal
-A_X$.calA_X$. These sections generate downstairs. Hence $mathcal A_X$calA_X$ is
+identifies the sections with those of the same power of mathcal
+A_X.calA_X. These sections generate downstairs. Hence mathcal A_XcalA_X is
globally generated after a sufficiently high multiple, so the
-divisor representing $mathcal A_X$,calA_X$, divided by $r$, is a semiample
-$Q$-divisor on $X$ numerically equivalent to $D$.
+divisor representing mathcal A_X,calA_X, divided by r, is a semiample
+Q-divisor on X numerically equivalent to D.
## The large-nef-adjoint model
-We may therefore replace $X$ by a small projective
-$Q$-factorialization and, at the end, descend the resulting
+We may therefore replace X by a small projective
+Q-factorialization and, at the end, descend the resulting
semiample divisor via Lemma~lemma:numsemiample-descent. For
-the rest of this section $X$ is klt and $Q$-factorial.
+the rest of this section X is klt and Q-factorial.
-Let $L$ be a positive multiple of $M$ that is Cartier. We write
-$L=m_0M$ and eventually absorb the factor $m_0$ into the nef
+Let L be a positive multiple of M that is Cartier. We write
+L=m_0M and eventually absorb the factor m_0 into the nef
summand.
-There are an integer $m_0>0$, a birational contraction
-$varphicolon Xdashrightarrow Y$ between klt $Q$-factorial
-projective varieties, and a common smooth model $W$ with
-$pcolon Wto X$ and $qcolon Wto Y$, satisfying the following
+There are an integer m_0>0, a birational contraction
+varphicolon Xdashrightarrow Y between klt Q-factorial
+projective varieties, and a common smooth model W with
+pcolon Wto X and qcolon Wto Y, satisfying the following
properties.
-
-item The transform $T'$ of $T=K_X+B$ is pseudo-effective, $N$ (the
-transform of $M$) is nef, and $T'+m_0 N$ is nef on $Y$.
-item The contraction $varphi$ is $M$-trivial: on $W$,
-$p^*M=q^*N$.
-item There is an effective $q$-exceptional rational divisor $Ege0$
-on $W$ with $p^*T=q^*T'+E$.
+ The transform T' of T=K_X+B is pseudo-effective, N (the
+transform of M) is nef, and T'+m_0 N is nef on Y.
+ The contraction varphi is M-trivial: on W,
+p^*M=q^*N.
+ There is an effective q-exceptional rational divisor Ege0
+on W with p^*T=q^*T'+E.
Consequently,
-$p^*(T+M)=q^*(T'+N)+E$ and, by
+p^*(T+M)=q^*(T'+N)+E and, by
Lemma~lemma:positive-properties(i),
[
P_sigma(p^*(T+M))=P_sigma(q^*(T'+N))
quadas divisors on W.
]
-So it suffices to prove (Z$_n$) for the pair $(Y,B_Y)$ with nef
-divisor $N$, and the adjoint $T'+m_0N$ is nef.
+So it suffices to prove (Z_n) for the pair (Y,B_Y) with nef
+divisor N, and the adjoint T'+m_0N is nef.
Proof.
-Apply Theorem~input:models to the generalized klt $Q$-pair
-$(X,B+m_0M)$ with nef data $m_0M$. Its generalized adjoint is
-$T+m_0M$, which is pseudo-effective since $T$ is pseudo-effective
-and $m_0M$ is nef. The theorem gives a terminating generalized
-program; we verify $M$-triviality.
+Apply Theorem~input:models to the generalized klt Q-pair
+(X,B+m_0M) with nef data m_0M. Its generalized adjoint is
+T+m_0M, which is pseudo-effective since T is pseudo-effective
+and m_0M is nef. The theorem gives a terminating generalized
+program; we verify M-triviality.
-Consider a $(T+m_0M)$-negative extremal ray, and let $C$ be a
+Consider a (T+m_0M)-negative extremal ray, and let C be a
rational curve generating it. By the length bound
-(Lemma~lemma:generalized-ray-bound), we may choose $C$ with
-$0<-Tcdot Cle2n$. If $Mcdot C>0$, then $m_0Mcdot Cge1$ by
-integrality (the divisor $m_0M$ is Cartier). Hence
+(Lemma~lemma:generalized-ray-bound), we may choose C with
+0<-Tcdot Cle2n. If Mcdot C>0, then m_0Mcdot Cge1 by
+integrality (the divisor m_0M is Cartier). Hence
[
(T+m_0M)cdot Cge -2n +1>0,
]
-a contradiction if $m_0>2n$. So $Mcdot C=0$ on every
-$(T+m_0M)$-negative extremal ray, and $M$ therefore descends
+a contradiction if m_0>2n. So Mcdot C=0 on every
+(T+m_0M)-negative extremal ray, and M therefore descends
through the contraction by Cartier descent in the ordinary cone
and contraction theorem. The same reasoning applies to a small
-flip, whose flipped curves are $M$-trivial by the same argument as
-for $(T+m_0M)$-negative rays. Effectively, every step is
-$M$-trivial.
+flip, whose flipped curves are M-trivial by the same argument as
+for (T+m_0M)-negative rays. Effectively, every step is
+M-trivial.
-Let $Y$ be the output. Nefness of $T'+m_0N$ is the output
-nefness of the program, and $T'$ pseudo-effectivity is
-preservation along the program. The $M$-triviality gives
-$p^*M=q^*N$ on $W$. The ordinary adjoint comparison of the
-program, applied to the displayed adjoint $K_X+B+m_0M$, gives
-$p^*(T+m_0M)=q^*(T'+m_0N)+E_{m_0}$ with $E_{m_0}$ effective
-$q$-exceptional; subtracting the $M$-pullback comparison
-$p^*(m_0M)=q^*(m_0N)$ yields $p^*T=q^*T'+E_{m_0}$. Setting
-$E=E_{m_0}$ gives the effective $q$-exceptional comparison.
+Let Y be the output. Nefness of T'+m_0N is the output
+nefness of the program, and T' pseudo-effectivity is
+preservation along the program. The M-triviality gives
+p^*M=q^*N on W. The ordinary adjoint comparison of the
+program, applied to the displayed adjoint K_X+B+m_0M, gives
+p^*(T+m_0M)=q^*(T'+m_0N)+E_{m_0} with E_{m_0} effective
+q-exceptional; subtracting the M-pullback comparison
+p^*(m_0M)=q^*(m_0N) yields p^*T=q^*T'+E_{m_0}. Setting
+E=E_{m_0} gives the effective q-exceptional comparison.
-After Lemma~lemma:large-nef-adjoint, we may replace $(X,B,M)$
-by $(Y,B_Y,N)$ and assume in addition that $T+mL$ is nef for some
-integer $m>0$.
+After Lemma~lemma:large-nef-adjoint, we may replace (X,B,M)
+by (Y,B_Y,N) and assume in addition that T+mL is nef for some
+integer m>0.
## The case of full nef dimension
-Suppose $T+mL$ has full nef dimension. Then
-(eq:full-condition) holds with $a=m$, and (P$_n$) shows that
-$T+M$ is big. Write, by Kodaira's lemma,
+Suppose T+mL has full nef dimension. Then
+(eq:full-condition) holds with a=m, and (P_n) shows that
+T+M is big. Write, by Kodaira's lemma,
[
T+Msim_Q A_1+G_1,
]
-where $A_1$ is ample rational and $G_1ge0$ is rational.
+where A_1 is ample rational and G_1ge0 is rational.
-Choose a rational $v>0$ sufficiently small that $(X,B+vG_1)$ is
-klt. The divisor $M+vA_1$ is ample (sum of a nef and an ample
-divisor). A general effective rational representative $A'$ of its
+Choose a rational v>0 sufficiently small that (X,B+vG_1) is
+klt. The divisor M+vA_1 is ample (sum of a nef and an ample
+divisor). A general effective rational representative A' of its
class may be chosen with all coefficients small enough that
-$(X,B+vG_1+A')$ remains klt. Its adjoint is
+(X,B+vG_1+A') remains klt. Its adjoint is
K_X+B+vG_1+A'sim_Q(1+v)(T+M),
-because $T+M=K_X+B+Msim_Q A_1+G_1$. This adjoint is big, since
-$T+M$ is big.
+because T+M=K_X+B+Msim_Q A_1+G_1. This adjoint is big, since
+T+M is big.
-The pair $(X,B+vG_1+A')$ is klt with big adjoint, so it has a good
+The pair (X,B+vG_1+A') is klt with big adjoint, so it has a good
minimal model by the big case of [cite]; equivalently, apply
Theorems~input:models and~input:abundance. On a common
-smooth resolution $w$, the left side of
+smooth resolution w, the left side of
(eq:big-ordinary-boundary) is the pullback of a semiample
divisor plus an effective exceptional divisor, by the effective
exceptional comparison of the program. Its positive part is
therefore the semiample pullback.
-Since $P_sigma$ is homogeneous,
+Since P_sigma is homogeneous,
[
P_sigma(w^*(T+M))
=frac1{1+v}P_sigma(w^*(K_X+B+vG_1+A'))
]
is a rational divisor numerically equivalent to a semiample
-$Q$-divisor on $w$. This is (Z$_n$) in the full-nef dimension
+Q-divisor on w. This is (Z_n) in the full-nef dimension
case.
## Descent along a nef reduction
-Now suppose $T+mL$ does not have full nef dimension. Let
-$hcolon Xdashrightarrow Z_1$ be the nef reduction of $T+mL$
+Now suppose T+mL does not have full nef dimension. Let
+hcolon Xdashrightarrow Z_1 be the nef reduction of T+mL
(Definition at the start of Ssec:preliminaries), so
-$dim Z_1=r<n$. Fix a general compact fibre $F$.
+dim Z_1=r<n. Fix a general compact fibre F.
-After shrinking $Z_1$, $F$ is klt and the pair $(F,B|_F)$ is klt
-with adjoint $T|_F=K_F+B|_F$. Since $(T+mL)|_{widetilde F}equiv0$
-on any component $widetilde F$ of the fibre, $T|_{widetilde F}$
+After shrinking Z_1, F is klt and the pair (F,B|_F) is klt
+with adjoint T|_F=K_F+B|_F. Since (T+mL)|_{widetilde F}equiv0
+on any component widetilde F of the fibre, T|_{widetilde F}
is numerically trivial and therefore nef. By
-Theorem~input:abundance, $T|_F$ is semiample; a semiample
+Theorem~input:abundance, T|_F is semiample; a semiample
numerically trivial divisor has zero-dimensional image, so
K_F+B|_Fsim_Q0.
We use the birational descent of nef divisors
-cite[Lemma~3.1]{LP}. After resolving $h$ and modifying its base
+cite[Lemma~3.1]{LP}. After resolving h and modifying its base
birationally, it supplies a diagram
[
Xxleftarrow{p}X_1xrightarrow{f}Z_1
]
-in which $X_1,Z_1$ are smooth and projective, $p$ is birational,
-$f$ is a contraction, and there is a rational divisor $M_1$ on
-$Z_1$ with
+in which X_1,Z_1 are smooth and projective, p is birational,
+f is a contraction, and there is a rational divisor M_1 on
+Z_1 with
p^*Mequiv f^*M_1.
-The divisor $M_1$ is nef: its pullback under a surjective morphism
-is numerically equivalent to the nef divisor $p^*M$, and nefness
+The divisor M_1 is nef: its pullback under a surjective morphism
+is numerically equivalent to the nef divisor p^*M, and nefness
descends under a surjective morphism.
-Write the crepant boundary as $B_1^+-B_1^-$ with effective rational
+Write the crepant boundary as B_1^+-B_1^- with effective rational
divisors of disjoint support, so that
-$K_{X_1}+B_1^+-B_1^-=p^*(K_X+B)$. Set $B_1=B_1^+$ and $E=B_1^-$.
-Since $B$ is effective, $E$ is $p$-exceptional. Klt singularities
-make all coefficients of $B_1$ less than one; after further
+K_{X_1}+B_1^+-B_1^-=p^*(K_X+B). Set B_1=B_1^+ and E=B_1^-.
+Since B is effective, E is p-exceptional. Klt singularities
+make all coefficients of B_1 less than one; after further
shrinking, its support is simple normal crossing. Thus
-$(X_1,B_1)$ is klt and
+(X_1,B_1) is klt and
- K_{X_1}+B_1=p^*T+E,qquad Ege0.
+ K_{X_1}+B_1=p^*T+E, Ege0.
-In particular the adjoint of $(X_1,B_1)$ is pseudo-effective.
+In particular the adjoint of (X_1,B_1) is pseudo-effective.
-A general fibre $F_1$ of $f$ maps birationally onto the
-corresponding compact fibre $F$ of the nef reduction. Every
-component of $E$ that meets $F_1$ restricts to an exceptional
-divisor over $F$. Equations~(eq:fibre-adjoint-trivial)
+A general fibre F_1 of f maps birationally onto the
+corresponding compact fibre F of the nef reduction. Every
+component of E that meets F_1 restricts to an exceptional
+divisor over F. Equations~(eq:fibre-adjoint-trivial)
and~(eq:resolved-boundary) therefore give
kappa(F_1,(K_{X_1}+B_1)|_{F_1})=0.
The restriction is an effective exceptional divisor up to rational
linear equivalence, and its section spaces are those of the trivial
-divisor on the normal fibre $F$. In particular the general fibre
+divisor on the normal fibre F. In particular the general fibre
pair has pseudo-effective adjoint and, by
Theorems~input:models and~input:abundance, a good
minimal model.
By (eq:nef-descent) we may replace the nef term by
-$f^*M_1$ before making positive-part comparisons; this pullback is
+f^*M_1 before making positive-part comparisons; this pullback is
trivial on every step of the relative program.
Let
[
X_2xrightarrow{g}Z_2xrightarrow{j}Z_1
]
-be the relative semiample fibration of $K_{X_2}+B_2$, with $g$ a
-contraction. Its construction gives a $j$-ample $Q$-Cartier
-divisor $A_2$ on $Z_2$ such that
+be the relative semiample fibration of K_{X_2}+B_2, with g a
+contraction. Its construction gives a j-ample Q-Cartier
+divisor A_2 on Z_2 such that
K_{X_2}+B_2sim_Q g^*A_2.
@@ -3104,84 +3088,84 @@ dimension is zero, and hence
Apply Ambro's canonical bundle formula consequence
cite[Theorem~0.2]{Ambro} to (eq:relative-iitaka). Its
-hypotheses hold: $(X_2,B_2)$ is a projective klt pair with effective
-rational boundary, $g$ is a contraction to a normal projective
+hypotheses hold: (X_2,B_2) is a projective klt pair with effective
+rational boundary, g is a contraction to a normal projective
variety, and the ordinary adjoint is rationally linearly pulled back
-from a $Q$-Cartier divisor. It gives an effective rational
-boundary $B_{Z_2}$ such that $(Z_2,B_{Z_2})$ is klt and
+from a Q-Cartier divisor. It gives an effective rational
+boundary B_{Z_2} such that (Z_2,B_{Z_2}) is klt and
K_{X_2}+B_2sim_Q g^*(K_{Z_2}+B_{Z_2}).
The base adjoint is pseudo-effective. One direct verification,
using only the inputs already in force, is the following. The
-pseudo-effective ordinary adjoint on $X_2$ has a good minimal model
+pseudo-effective ordinary adjoint on X_2 has a good minimal model
by Theorems~input:models and~input:abundance, so it has
a nonzero section in some positive multiple. Since
-$g_*OO_{X_2}=OO_{Z_2}$, the projection formula and
+g_*OO_{X_2}=OO_{Z_2}, the projection formula and
(eq:base-adjoint) identify this section space with the
-corresponding section space on $Z_2$. Thus $K_{Z_2}+B_{Z_2}$ even
+corresponding section space on Z_2. Thus K_{Z_2}+B_{Z_2} even
has an effective rationally linearly equivalent representative.
-The pair $(Z_2,B_{Z_2})$ is klt of dimension $dim Z_2<n$. Let
-$M_{Z_2}=j^*M_1$ be the pullback of $M_1$ to $Z_2$: it is nef
-(random check: $j$ is a morphism so $j^*M_1$ is nef). The
-induction hypothesis (Z$_{dim Z_2}$) applies and gives a smooth
-model $ncolon W_{Z_2}to Z_2$ with
+The pair (Z_2,B_{Z_2}) is klt of dimension dim Z_2<n. Let
+M_{Z_2}=j^*M_1 be the pullback of M_1 to Z_2: it is nef
+(random check: j is a morphism so j^*M_1 is nef). The
+induction hypothesis (Z_{dim Z_2}) applies and gives a smooth
+model ncolon W_{Z_2}to Z_2 with
[
P_sigma(n^*(K_{Z_2}+B_{Z_2}+M_{Z_2}))
]
-rational and numerically equivalent to a semiample $Q$-divisor on
-$W_{Z_2}$.
+rational and numerically equivalent to a semiample Q-divisor on
+W_{Z_2}.
-Pull this back. Let $g'colon W_2to X_2$ be a smooth model onto
-which $W_{Z_2}$ pulls back via $g$, so that
+Pull this back. Let g'colon W_2to X_2 be a smooth model onto
+which W_{Z_2} pulls back via g, so that
[
g'^*(K_{X_2}+B_2+M_2)
=g'^*g^*(K_{Z_2}+B_{Z_2}+M_{Z_2}),
]
-where $M_2=g^*M_{Z_2}=g^*j^*M_1$. By
+where M_2=g^*M_{Z_2}=g^*j^*M_1. By
Lemma~lemma:positive-properties(ii),
[
P_sigma(g'^*(K_{X_2}+B_2+M_2))
=g'^*P_sigma(n^*(K_{Z_2}+B_{Z_2}+M_{Z_2}))
]
is a rational divisor numerically equivalent to a semiample
-$Q$-divisor on $W_2$.
+Q-divisor on W_2.
-Transfer to $X$: on $W_2$, the pullback
-$p_{new}^*(T+M)$ of the original adjoint sum, under the
-composition $p_{new}=pcirc(X_2to X_1)circ(X_2to X_1)$
-of birational morphisms, differs from $K_{X_2}+B_2+M_2$ by an
-effective $p_{new}$-exceptional divisor $E_2$. By
+Transfer to X: on W_2, the pullback
+p_{new}^*(T+M) of the original adjoint sum, under the
+composition p_{new}=pcirc(X_2to X_1)circ(X_2to X_1)
+of birational morphisms, differs from K_{X_2}+B_2+M_2 by an
+effective p_{new}-exceptional divisor E_2. By
Lemma~lemma:positive-properties(i),
[
P_sigma(p_{new}^*(T+M))
=P_sigma(K_{X_2}+B_2+M_2+E_2)
=P_sigma(K_{X_2}+B_2+M_2),
]
-which we have shown to be numerially semiample. Hence (Z$_n$)
+which we have shown to be numerially semiample. Hence (Z_n)
holds in the nef-reduction case.
-Combining the full-nef and nef-reduction cases, (Z$_n$) holds for
-every $(X,B,M)$ satisfying the hypotheses of (Z$_n$). This closes
+Combining the full-nef and nef-reduction cases, (Z_n) holds for
+every (X,B,M) satisfying the hypotheses of (Z_n). This closes
both simultaneous inductions and completes the proof of
-Theorem~thm:key-ingredient for every $n$ via
+Theorem~thm:key-ingredient for every n via
Section~sec:completion.
-## Proof of the main theorem over $C$
+## Proof of the main theorem over C
With both inductions established, we can now prove
-Theorem~thm:main over $C$. Let $(X,B)$ be a projective klt
-$Q$-pair of dimension $n$, $M$ a nef $Q$-Cartier divisor, and
-$K_X+B$ pseudo-effective. The sum $D=K_X+B+M$ is nef by
-hypothesis. On any smooth birational model $wcolon Wto X$,
+Theorem~thm:main over C. Let (X,B) be a projective klt
+Q-pair of dimension n, M a nef Q-Cartier divisor, and
+K_X+B pseudo-effective. The sum D=K_X+B+M is nef by
+hypothesis. On any smooth birational model wcolon Wto X,
[
P_sigma(w^*D)=w^*D,
]
-because a nef divisor has zero negative part. Statement (Z$_n$)
+because a nef divisor has zero negative part. Statement (Z_n)
applies and gives that this pullback is rational and numerically
-equivalent to a semiample $Q$-divisor on $W$. Since a klt variety
+equivalent to a semiample Q-divisor on W. Since a klt variety
has rational singularities,
Lemma~lemma:numsemiample-descent descends the result to a
-semiample $Q$-Cartier divisor on $X$ that is numerically equivalent
-to $D$. This is Theorem~thm:main over $C$.
+semiample Q-Cartier divisor on X that is numerically equivalent
+to D. This is Theorem~thm:main over C.