diff b1f9bc4..56bc728 log_abundance_char_zero_v2.txt
diff --git a/log_abundance_char_zero_v2.txt b/log_abundance_char_zero_v2.txt
index 1dc012e..3b62f1a 100644
--- a/log_abundance_char_zero_v2.txt
+++ b/log_abundance_char_zero_v2.txt
@@ -60,9 +60,9 @@ maketitle
Abstract:
We prove the rational-boundary log abundance conjecture in every
dimension over algebraically closed fields of characteristic zero:
-every nef $Q$-Cartier log canonical divisor on a projective log
+every nef Q-Cartier log canonical divisor on a projective log
canonical pair with effective rational boundary is semiample. Over
-$C$, the same proof establishes smooth canonical nonvanishing in
+C, the same proof establishes smooth canonical nonvanishing in
every dimension. The proof is a dimension induction; the hard input
is nonvanishing, closed by a reduction to positive characteristic and
a determinant argument. It is independent of logarithmic Iitaka
@@ -80,58 +80,57 @@ addcontentsline{toc}{section}{A guide for the reader}
## The statement we prove
-The log abundance conjecture says this: if $K_X+B$ meets every
-curve of $X$ nonnegatively, then some positive multiple of $K_X+B$ is
+The log abundance conjecture says this: if K_X+B meets every
+curve of X nonnegatively, then some positive multiple of K_X+B is
a genuine line bundle generated by its global sections. Nefness is a
numerical condition on curve degrees. Generation by sections is an
algebraic condition on actual functions. Log abundance says the first
forces the second.
[Log abundance, main theorem]
-Let $(X,B)$ be a normal projective log canonical pair over an
-algebraically closed field of characteristic zero, with $B$ an
-effective rational divisor and $K_X+B$ $Q$-Cartier. If $K_X+B$ is
-nef, then some positive Cartier multiple of $K_X+B$ is globally
+Let (X,B) be a normal projective log canonical pair over an
+algebraically closed field of characteristic zero, with B an
+effective rational divisor and K_X+B Q-Cartier. If K_X+B is
+nef, then some positive Cartier multiple of K_X+B is globally
generated.
-The body of the paper works over $C$. The passage to an arbitrary
+The body of the paper works over C. The passage to an arbitrary
algebraically closed field of characteristic zero is bookkeeping and
lives in Section~sec:completion.
## The shape of the argument
-The proof is a dimension induction. In dimension $n$ it aims to
+The proof is a dimension induction. In dimension n it aims to
produce two outputs, in this order.
[label=(roman*)]
-item Smooth canonical nonvanishing (NV$_n$). A smooth
-projective complex $n$-fold $X$ with pseudo-effective $K_X$ has
-$H^0(X,mK_X)neq0$ for some $m>0$.
-
-item Good log minimal models in dimension $n$ for all
+ Smooth canonical nonvanishing (NV_n). A smooth
+projective complex n-fold X with pseudo-effective K_X has
+H^0(X,mK_X)neq0 for some m>0.
+ Good log minimal models in dimension n for all
projective lc pairs with effective real boundary and pseudo-effective
adjoint. On a good model the adjoint is semiample. This is the
inductive assertion; it yields Theorem~thm:main in dimension
-$n$ by comparing the pair with its good model.
+n by comparing the pair with its good model.
-The two outputs are entangled. (ii) in dimension $n-1$ (the induction
-hypothesis) is used to prove (i) in dimension $n$. And (i) in
-dimension $n$, together with the signed-representative theorem
+The two outputs are entangled. (ii) in dimension n-1 (the induction
+hypothesis) is used to prove (i) in dimension n. And (i) in
+dimension n, together with the signed-representative theorem
(Sections~sg:section--sg:descent), is used to prove
-(ii) in dimension $n$. Together, (i)+(ii) in dimension $n$ supply the
+(ii) in dimension n. Together, (i)+(ii) in dimension n supply the
induction hypothesis for the next dimension. The base case is a point.
The engine that turns nonvanishing into abundance is the
signed-representative theorem (Theorem~thm:signed). Its
-statement is subtle: it handles a nef $L=K_X+D$ on a $Q$-factorial
-dlt pair, with $D$ reduced and with $L$ rational-linearly equivalent
-to a $Q$-divisor with either positive or negative coefficients on
-the components of $D$, provided
-(i) $L-cD$ is pseudo-effective for some $c>0$, and
-(ii) $L|_D$ is semiample. Under those hypotheses it produces abundance:
-$kappa(L)=nu(L)$. The signed-representative theorem is what turns
+statement is subtle: it handles a nef L=K_X+D on a Q-factorial
+dlt pair, with D reduced and with L rational-linearly equivalent
+to a Q-divisor with either positive or negative coefficients on
+the components of D, provided
+(i) L-cD is pseudo-effective for some c>0, and
+(ii) L|_D is semiample. Under those hypotheses it produces abundance:
+kappa(L)=nu(L). The signed-representative theorem is what turns
a numerically-nonnegative adjoint into a genuinely semiample one.
Its proof (Sections~sg:section--sg:descent) uses roots,
Hodge modules, and a formal deformation argument; it is the technical
@@ -140,24 +139,24 @@ heart of the paper.
## Where the difficulty lives
Smooth canonical nonvanishing is the bottleneck. Everything else in
-the induction is formal once NV$_n$ is in hand:
+the induction is formal once NV_n is in hand:
Section~sec:induction proves
Proposition~prop:inductive-reduction, which makes good models of
-$(X,Delta)$ for real boundary once NV$_n$ holds. The signed-
+(X,Delta) for real boundary once NV_n holds. The signed-
representative theorem fills the zero-Iitaka case; the case of
positive Kodaira dimension is Fujino--Gongyo. Everything rests on
-NV$_n$.
+NV_n.
-The proof of NV$_n$ assumes a counterexample and derives a
+The proof of NV_n assumes a counterexample and derives a
contradiction in four stages. This is the pivotal movement of the
paper, and the earlier part of our reading guide has no more to say
about it beyond a signpost to the relevant sections.
medskipnoindent
Stage 1: geometric exclusions (Section~sec:exclusions).
-The counterexample $X$ must:
+The counterexample X must:
(a) have irregularity zero, so numerical equivalence equals
-$Q$-linear equivalence on its rational divisors;
+Q-linear equivalence on its rational divisors;
(b) not be covered by proper subvarieties of non-general type;
(c) admit no positive canonical current annihilating the fibers of a
rational fibration (this rules out families of subvarieties pulled
@@ -173,12 +172,12 @@ theorem applied to a positive canonical current.
medskipnoindent
Stage 2: two opposing jet estimates (Sections~sec:common-tools
-and~sec:jets). At a very general point of $X$, every section of
-every Cartier multiple of a suitable normalized nef divisor $P$ has
+and~sec:jets). At a very general point of X, every section of
+every Cartier multiple of a suitable normalized nef divisor P has
bounded vanishing order (the scalar jet bound). But on the
-projective bundle over $Xtimes X$ associated to two copies of the
+projective bundle over Xtimes X associated to two copies of the
same line bundle, the very general Seshadri constant exceeds
-$2n+2$, so jets vanish to very high order (the two-slot
+2n+2, so jets vanish to very high order (the two-slot
separation). The first bound consumes the curve exclusion (d); the
second is what produces the determinant incompatibility of Stage 3.
@@ -195,7 +194,7 @@ rank deficit bounds it from below. Contradiction for large primes.
medskipnoindent
Stage 4: contradiction (Section~fr:section, end).
-The counterexample was impossible. NV$_n$ holds.
+The counterexample was impossible. NV_n holds.
## Reading paths
@@ -226,16 +225,16 @@ The full proof, end to end. Read linearly.
The proof uses three external inputs:
[label=(alph*)]
-item Logarithmic Iitaka subadditivity, stated below as
+ Logarithmic Iitaka subadditivity, stated below as
Theorem~asm:iitaka. It is used once, in
Lemma~ex:irregularity, and nowhere else. The zero-boundary
case used there also follows from Hacon--Popa--Schnell and from
Cao--Pu{a}un.
-item Good minimal models of big klt adjoints, from Birkar--Cascini--
+ Good minimal models of big klt adjoints, from Birkar--Cascini--
Hacon--McKernan [cite]. These are used in the signed-
representative engine and in the reduction at positive Kodaira
dimension.
-item Ordinary negative-ample Kodaira--Saito vanishing and the
+ Ordinary negative-ample Kodaira--Saito vanishing and the
Du~Bois machinery of Saito [cite], used in the
Hodge step of the engine.
@@ -246,10 +245,10 @@ finite-generation corollary, after abundance has been proved.
## Notation
-$(X,B)$ always denotes a log canonical pair with $B$ effective,
+(X,B) always denotes a log canonical pair with B effective,
rational boundary unless stated otherwise. ``Nef'' means nonnegative
on every curve. ``Semiample'' means a positive multiple is generated
-by global sections. ``Abundant'' means $kappa(L)=nu(L)$, i.e.~the
+by global sections. ``Abundant'' means kappa(L)=nu(L), i.e.~the
Iitaka dimension equals the numerical dimension. Real-boundary
statements occur only in the induction (Section~sec:induction);
the main theorem has rational boundary. All other conventions are
@@ -261,8 +260,8 @@ newpage
# Introduction
%======================================================================
-Let $(X,B)$ be a projective log canonical pair with rational boundary.
-Its log canonical divisor $K_X+B$ packages the canonical class and
+Let (X,B) be a projective log canonical pair with rational boundary.
+Its log canonical divisor K_X+B packages the canonical class and
the boundary. A rational Cartier divisor is nef if its degree
on every curve is nonnegative; it is semiample if a positive
Cartier multiple is generated by global sections. Such a multiple
@@ -270,7 +269,7 @@ defines a morphism, so semiampleness turns numerical positivity into a
fibration described by sections. Log abundance asserts that semiampleness
holds whenever nefness does (Theorem~thm:main).
-The proof runs by induction on $dim X$. Its inductive hypothesis is
+The proof runs by induction on dim X. Its inductive hypothesis is
stronger than the main theorem: every projective lc pair with effective
real boundary and pseudo-effective real Cartier adjoint
has a good log minimal model, on which the adjoint is semiample.
@@ -279,40 +278,35 @@ canonical class on a smooth projective variety has a nonzero
pluricanonical section --- is proved at each dimension before
the good-model output of that dimension is used. At the very end, descent
of the evaluation map of an actual Cartier line bundle transfers the
-result over $C$ to every algebraically closed field of characteristic
+result over C to every algebraically closed field of characteristic
zero (Section~sec:completion).
-Throughout, $kappa(L)$ is the Iitaka dimension of a rational divisor
-$L$, with $kappa(L)=-∞$ if every positive integral system is
-empty; $nu(L)$ is the numerical dimension. For nef $L$ we always have
-$kappa(L)lenu(L)$; abundance is the reverse inequality.
+Throughout, kappa(L) is the Iitaka dimension of a rational divisor
+L, with kappa(L)=-∞ if every positive integral system is
+empty; nu(L) is the numerical dimension. For nef L we always have
+kappa(L)lenu(L); abundance is the reverse inequality.
## Consequences covered by this paper
The consequences of the main theorem fall into four groups, each with
its own section, each recording the exact hypotheses of the cited
external theorems it consumes.
-
-
-item Good models, linear triviality, slc, and dlt extension
-(Section~sec:consequences). Good models over $C$ for real
+ Good models, linear triviality, slc, and dlt extension
+(Section~sec:consequences). Good models over C for real
boundaries; the smooth-canonical good model; rational-linear
-triviality when $kappa=0$; abundance for semi-log-canonical pairs
+triviality when kappa=0; abundance for semi-log-canonical pairs
after normalization; supported extension of pluricanonical sections
across the reduced dlt boundary.
-
-item Finite generation of rounded log canonical rings
+ Finite generation of rounded log canonical rings
(Section~sec:canonical-rings). Both absolute (projective over
a field of characteristic zero) and relative (proper over an
-algebraic base over $C$).
-
-item Classification-flavoured consequences
+algebraic base over C).
+ Classification-flavoured consequences
(Section~sec:classification): rational connectedness via
Mumford's criterion; base of the MRC fibration via cotangent tensors;
-finiteness of $pi_1$ when $kappa=0$ and $chine0$; quasi-projective
+finiteness of pi_1 when kappa=0 and chine0; quasi-projective
universal covers in the sense of Claudon--H"oring--Koll'ar.
-
-item Relative analytic abundance (Section~sec:analytic-abundance):
+ Relative analytic abundance (Section~sec:analytic-abundance):
projective surjective morphisms of normal complex analytic varieties
with rational lc adjoint nef over the whole base, obtaining a relatively
generated Cartier multiple in a neighborhood of each compact base
@@ -345,16 +339,16 @@ cite[Corollary~1.1.2]{BCHM2010}. Two distinctions remain sharp on the
boundary of the positive cone.
[label=(alph*)]
-item Nonvanishing produces sections but not semiampleness.
+ Nonvanishing produces sections but not semiampleness.
Hashizume cite[Theorem~1.4]{Hashizume2018} reduces real-boundary lc
-nonvanishing and ordinary minimal-model existence in dimension $le n$
-to smooth canonical nonvanishing in dimension $n$. The induction must
+nonvanishing and ordinary minimal-model existence in dimension le n
+to smooth canonical nonvanishing in dimension n. The induction must
then make these models good. In positive Kodaira dimension the
required step is Fujino--Gongyo
cite[Lemma~3.5]{FujinoGongyoRings2014}.
-item Kodaira dimension zero and numerical dimension zero are distinct
+ Kodaira dimension zero and numerical dimension zero are distinct
hypotheses. Gongyo cite[Theorem~1.2]{Gongyo2013} proves that a
-numerically trivial rational lc adjoint is $Q$-linearly trivial. In
+numerically trivial rational lc adjoint is Q-linearly trivial. In
the zero-Iitaka argument below, numerical triviality forces an already
available effective representative to vanish.
@@ -374,30 +368,30 @@ the finite-generation part of Section~sec:canonical-rings.
## The two tasks in the induction
-Assume good models exist in all dimensions below $n$. We must prove
-(i)~NV$_n$ and (ii)~good models in dimension $n$. Both jobs use the
+Assume good models exist in all dimensions below n. We must prove
+(i)~NV_n and (ii)~good models in dimension n. Both jobs use the
same criterion for a signed representative on a reduced boundary,
which we state here once and then prove in Sections~sg:section--%
sg:descent.
-Let $L$ be a nef divisor and $nu(L)
-=max{j : L^jH^{n-j}>0}$ for $H$ ample and $n=dim X$; this is the
+Let L be a nef divisor and nu(L)
+=max{j : L^jH^{n-j}>0} for H ample and n=dim X; this is the
numerical dimension. The signed-representative criterion is
[
-L=K_X+D nef,qquad L-cD pseudo-effective for some c>0,
-Lsim_{Q}textstylesum_i a_iD_i,qquad L|_D semiample as a
+L=K_X+D nef, L-cD pseudo-effective for some c>0,
+Lsim_{Q}textstylesum_i a_iD_i, L|_D semiample as a
rational line bundle,
-⇒quad kappa(L)=nu(L).
+⇒ kappa(L)=nu(L).
]
-Here $(X,D)$ is a projective $Q$-factorial dlt pair, and $D=sum_iD_i$
+Here (X,D) is a projective Q-factorial dlt pair, and D=sum_iD_i
is the decomposition of the reduced boundary into prime components.
-The rational coefficients $a_i$ are allowed to be positive,
+The rational coefficients a_i are allowed to be positive,
negative, or zero. The hypothesis is semiampleness of
-$L|_D$ on the entire reduced scheme $D$, so normalisation and
+L|_D on the entire reduced scheme D, so normalisation and
conductor gluing are built in (they enter via
cite[Theorem~1.5]{FujinoGongyo2014}). For the good-model task, positive
Kodaira dimension is Fujino--Gongyo cite[Lemma~3.5]{FujinoGongyoRings2014};
@@ -405,20 +399,20 @@ the signed criterion handles the remaining zero-Iitaka case
(Proposition~prop:inductive-reduction).
The signed criterion is proved geometrically.
-A generated multiple of $L|_D$ gives a morphism from $D$ to projective
+A generated multiple of L|_D gives a morphism from D to projective
space. Over a general point of the image of a positive-coefficient
component, a root construction separates that component from the
negative and coefficient-zero boundary. A filtered Hodge-module
calculation lifts local parameters on this image, together with a
parameter transverse to the lifted boundary, through all infinitesimal
neighborhoods. Fixing the lifted base parameters and varying the
-transverse one deforms a boundary fiber off $D$ to a projective
+transverse one deforms a boundary fiber off D to a projective
subvariety. Its variations give a dominating family of subvarieties on
-which $L$ is numerically trivial. The dimension of this family is
-$n-nu(L)$, which bounds the base dimension of the nef reduction of
-$L$ from below by $nu(L)$; descent along the reduction then
-identifies $L$ as the pullback of a big divisor on the base, so
-$kappa(L)genu(L)$, hence abundance.
+which L is numerically trivial. The dimension of this family is
+n-nu(L), which bounds the base dimension of the nef reduction of
+L from below by nu(L); descent along the reduction then
+identifies L as the pullback of a big divisor on the base, so
+kappa(L)genu(L), hence abundance.
The formal-deformation shape has a direct precedent in Miyaoka's
construction, who credits Reid with the suggestion to analyse a
@@ -449,9 +443,9 @@ Section~sec:hodge carries out exactly this identification.
## Smooth nonvanishing
-The proof of NV$_n$ assumes a counterexample
-$X$ (smooth projective $n$-fold, $K_X$ pseudo-effective, $kappa(X,K_X)
-=-∞$) and proceeds through four stages.
+The proof of NV_n assumes a counterexample
+X (smooth projective n-fold, K_X pseudo-effective, kappa(X,K_X)
+=-∞) and proceeds through four stages.
Exclusions:
(Section~sec:exclusions). The zero-boundary case of logarithmic
@@ -459,13 +453,12 @@ Iitaka subadditivity (Theorem~asm:iitaka) rules out positive
irregularity via the Albanese map. This is its only use. Geometric
preparations then exclude three ways the counterexample could be
covered by lower-dimensional subvarieties:
-
-item families carrying a positive current pulled back from a
+ families carrying a positive current pulled back from a
smaller variety;
-item families of curves of uniformly bounded genus and uniformly
+ families of curves of uniformly bounded genus and uniformly
bounded degree, after normalising the nef divisors to a fixed small
volume;
-item families of birational correspondences between fixed finite
+ families of birational correspondences between fixed finite
covers whose graphs dominate the product.
The first two use positive currents, algebraic webs, and an ascending
@@ -480,11 +473,11 @@ Jets:
(Sections~sec:common-tools and~sec:jets). The exclusions
produce two incompatible estimates for vanishing at a moving point.
At a very general point, every nonzero section of every Cartier
-multiple $kP$ of a suitable normalised nef divisor $P$ has vanishing
-order bounded by $k$ times a small normalised constant. On the other
-hand, on the projective bundle over $Xtimes X$ associated to two
+multiple kP of a suitable normalised nef divisor P has vanishing
+order bounded by k times a small normalised constant. On the other
+hand, on the projective bundle over Xtimes X associated to two
copies of a line bundle, the Seshadri constant at a very general point
-exceeds $2n+2$, so jets of very high order exist. The two summands
+exceeds 2n+2, so jets of very high order exist. The two summands
let the argument move in either factor.
The first estimate fails only if the corresponding jet kernel has a
@@ -512,7 +505,7 @@ the rank by a factor of four. The determinant of any nonzero maximal
minor is a section of an external product of two line bundles. The
first estimate --- expressed by a fixed movable curve on a blowup ---
bounds the order of the determinant at the selected diagonal point.
-The rank deficit bounds the same order from below by $3/4$ of the rank.
+The rank deficit bounds the same order from below by 3/4 of the rank.
Contradiction for large primes.
Theorem~common:finite-data-frobenius states this incompatibility
@@ -547,22 +540,22 @@ application is confined to Lemma~ex:irregularity.
[Logarithmic Iitaka subadditivity,
cite[Corollary~6.2]{OpenAIIitaka2026}]
-Let $f:Xto Y$ be a surjective morphism with connected fibres between
-smooth connected projective complex varieties. Let $D_X,D_Y$ be
+Let f:Xto Y be a surjective morphism with connected fibres between
+smooth connected projective complex varieties. Let D_X,D_Y be
reduced effective SNC divisors, allowing zero boundaries, with
-$Supp(f^*D_Y)⊆Supp(D_X)$. For a very general smooth fibre
-$F$, put $D_F=D_X|_F$. Then
+Supp(f^*D_Y)⊆Supp(D_X). For a very general smooth fibre
+F, put D_F=D_X|_F. Then
[
kappa(X,K_X+D_X)gekappa(F,K_F+D_F)+kappa(Y,K_Y+D_Y).
]
-Here $D_F$ is reduced with simple normal crossings and
-$K_F+D_Fsim(K_X+D_X)|_F$. The convention
-$(-∞)+b=-∞$ also holds when $b=-∞$, and the zero
+Here D_F is reduced with simple normal crossings and
+K_F+D_Fsim(K_X+D_X)|_F. The convention
+(-∞)+b=-∞ also holds when b=-∞, and the zero
divisor on a point has Iitaka dimension zero.
-The support inclusion allows extra components in $D_X$; the morphism
-need not be smooth away from $D_X$. The theorem imposes no nefness,
+The support inclusion allows extra components in D_X; the morphism
+need not be smooth away from D_X. The theorem imposes no nefness,
abundance, bigness, or good-model condition. Its zero-boundary case is
used in the Albanese reduction of Section~sec:exclusions; this
is the only subadditivity input in the proof. The companion's
@@ -605,25 +598,25 @@ counterexample geometry.
## Conventions
A variety is integral. Canonical divisors are chosen compatibly
-on birational models. For a divisor $E$ on a smooth model $p:Wto X$,
+on birational models. For a divisor E on a smooth model p:Wto X,
the log discrepancy is
[
a(E;X,B)=1+ord_E(K_W-p^*(K_X+B)).
]
-Log canonical means $a(E;X,B)ge0$ for every divisor over $X$; klt
-means $a(E;X,B)>0$. We use the standard dlt, plt, and slc conventions.
+Log canonical means a(E;X,B)ge0 for every divisor over X; klt
+means a(E;X,B)>0. We use the standard dlt, plt, and slc conventions.
For non-nef pseudo-effective divisors, statements about numerical
-dimension use Nakayama's $kappa_sigma$ as specified in the relevant
+dimension use Nakayama's kappa_sigma as specified in the relevant
reduction; these are not silently interchanged with the
-intersection-theoretic numerical dimension $nu$.
+intersection-theoretic numerical dimension nu.
For a divisor with nonempty positive system, the Iitaka dimension is
the maximum dimension of the images of its positive systems under the
-associated rational maps. We write $equiv$ for numerical equivalence
-and $sim_{Q}$ for rational-linear equivalence.
+associated rational maps. We write equiv for numerical equivalence
+and sim_{Q} for rational-linear equivalence.
-Unless a different base field is specified, constructions are over $C$.
+Unless a different base field is specified, constructions are over C.
A very general point of a variety avoids a countable union of
proper closed subvarieties. The transfer to arbitrary algebraically
closed characteristic-zero fields is made in Section~sec:completion
@@ -634,29 +627,29 @@ only after the complex argument has been completed.
%======================================================================
The dimension induction has two outputs in each dimension: smooth
-canonical nonvanishing (NV$_n$) and good log minimal models for
+canonical nonvanishing (NV_n) and good log minimal models for
projective lc pairs with effective real boundary and pseudo-effective
adjoint. The present section proves the passage from the first output
to the second, given good models strictly below the current dimension.
-All varieties here are projective over $C$.
+All varieties here are projective over C.
## Definitions and the induction hypothesis
-For an effective real boundary $Delta$, a good log minimal
-model of $(X,Delta)$ is a log minimal model on which the adjoint
+For an effective real boundary Delta, a good log minimal
+model of (X,Delta) is a log minimal model on which the adjoint
divisor is semiample. For a real boundary, semiampleness means
real-linear equivalence to the pullback of an ample real divisor by a
contraction. We use the standard definition of log minimal model in
which extracted prime divisors are included with coefficient one.
-For a pseudo-effective divisor $A$ we write $kappa_sigma(X,A)$ for
-Nakayama's numerical dimension. When $A$ is nef it agrees with the
-intersection-theoretic numerical dimension $nu(X,A)$.
+For a pseudo-effective divisor A we write kappa_sigma(X,A) for
+Nakayama's numerical dimension. When A is nef it agrees with the
+intersection-theoretic numerical dimension nu(X,A).
-Fix an integer $nge1$.
+Fix an integer nge1.
[Lower-dimensional good models]
-Every projective log canonical pair of dimension $<n$, with effective
+Every projective log canonical pair of dimension <n, with effective
real boundary and pseudo-effective adjoint divisor, has a good log
minimal model.
@@ -666,25 +659,25 @@ Assumption~mmp:lower-models. The induction runs as follows.
small
{@{}p{0.13textwidth}p{0.40textwidth}p{0.38textwidth}@{}}
-toprule
+
Stage & Hypothesis & Output and location
-midrule
-Base & Dimension $0$ & $GLM_{le0}$: a point
+
+Base & Dimension 0 & GLM_{le0}: a point
Smooth step & Assumption~mmp:lower-models and
Theorem~asm:iitaka &
- NV$_n$, Theorem~thm:smooth-nonvanishing
-Good-model step & Assumption~mmp:lower-models and NV$_n$ &
- $GLM_{le n}$,
+ NV_n, Theorem~thm:smooth-nonvanishing
+Good-model step & Assumption~mmp:lower-models and NV_n &
+ GLM_{le n},
Proposition~prop:inductive-reduction
-Nef conclusion & $GLM_{le n}$ &
- Semiampleness in dimension $n$, Lemma~mmp:comparison
-bottomrule
+Nef conclusion & GLM_{le n} &
+ Semiampleness in dimension n, Lemma~mmp:comparison
-Here $GLM_{le d}$ denotes good-model existence for all
+
+Here GLM_{le d} denotes good-model existence for all
projective complex lc pairs with effective real boundary and
-pseudo-effective adjoint in dimensions at most $d$; NV$_n$ and
+pseudo-effective adjoint in dimensions at most d; NV_n and
Semiampleness are the two other named outputs. The signed-
representative theorem used in both rows is proved in
Sections~sg:section--sg:descent; its hypothesis of
@@ -694,30 +687,30 @@ Proposition~prop:boundary-restriction.
## Comparison with a nef model
[Comparison with a nef model]
-Let $(X,Delta)$ be log canonical, and let $(X',Delta')$ be a log
+Let (X,Delta) be log canonical, and let (X',Delta') be a log
minimal model. On a common resolution
[
Xxleftarrow{u}Wxrightarrow{v}X',
]
-there is an effective $v$-exceptional divisor $F$ with
+there is an effective v-exceptional divisor F with
u^*(K_X+Delta)=v^*(K_{X'}+Delta')+F.
-If $K_X+Delta$ is nef, then $F=0$. Consequently, a nef rational
+If K_X+Delta is nef, then F=0. Consequently, a nef rational
adjoint is semiample whenever it has a good log minimal model. If
-$(X,Delta)$ is klt, then so is its log minimal model, and the model
+(X,Delta) is klt, then so is its log minimal model, and the model
extracts no divisors.
Proof.
-Put $F=u^*(K_X+Delta)-v^*(K_{X'}+Delta')$. Discrepancy improvement
-on divisors of $X$ gives $u_*Fge0$, while $-F$ is $u$-nef because
-$K_{X'}+Delta'$ is nef. The negativity lemma gives $Fge0$. At a
-prime of $W$ not $v$-exceptional, the coefficient of $F$ is zero unless
-that prime is extracted on $X'$. In the latter case it equals the
-negative of the log discrepancy over $(X,Delta)$; it is therefore
-nonpositive and so zero. Hence $F$ is $v$-exceptional. If the source
-adjoint is nef, $F$ is also $v$-nef, and negativity gives $F=0$.
+Put F=u^*(K_X+Delta)-v^*(K_{X'}+Delta'). Discrepancy improvement
+on divisors of X gives u_*Fge0, while -F is u-nef because
+K_{X'}+Delta' is nef. The negativity lemma gives Fge0. At a
+prime of W not v-exceptional, the coefficient of F is zero unless
+that prime is extracted on X'. In the latter case it equals the
+negative of the log discrepancy over (X,Delta); it is therefore
+nonpositive and so zero. Hence F is v-exceptional. If the source
+adjoint is nef, F is also v-nef, and negativity gives F=0.
The equality of pullbacks transfers semiampleness of a rational adjoint.
Indeed a proper birational morphism onto a normal variety has direct
@@ -734,29 +727,29 @@ inequalities make the model klt.
[Semiampleness on the reduced boundary]
-Assume Assumption~mmp:lower-models. Let $(V,D)$ be a projective
-$Q$-factorial dlt $n$-fold with $D$ reduced and $M=K_V+D$ nef. Then
-$M|_D$ is semiample as a rational line bundle on the reduced scheme $D$.
+Assume Assumption~mmp:lower-models. Let (V,D) be a projective
+Q-factorial dlt n-fold with D reduced and M=K_V+D nef. Then
+M|_D is semiample as a rational line bundle on the reduced scheme D.
Proof.
-The ambient $V$ is klt. The reduced floor of a $Q$-factorial dlt pair
-satisfies Serre's condition $S_2$ and has ordinary double crossings in
+The ambient V is klt. The reduced floor of a Q-factorial dlt pair
+satisfies Serre's condition S_2 and has ordinary double crossings in
codimension one; its irreducible components are normal.
-For the $S_2$ assertion, work locally with the cyclic class cover of
-the $Q$-Cartier divisor $D$. This cover is quasi-'etale and klt, hence
-Cohen--Macaulay. The eigensheaf $OO_V(-D)$ is a direct summand of the
+For the S_2 assertion, work locally with the cyclic class cover of
+the Q-Cartier divisor D. This cover is quasi-'etale and klt, hence
+Cohen--Macaulay. The eigensheaf OO_V(-D) is a direct summand of the
finite direct image, hence Cohen--Macaulay. The divisor sequence then
-shows that $D$ itself is Cohen--Macaulay.
+shows that D itself is Cohen--Macaulay.
-Divisorial adjunction on the normalisation $coprod D_ito D$ gives lc
-pairs $(D_i,Diff_{D_i}(D-D_i))$, where the different
+Divisorial adjunction on the normalisation coprod D_ito D gives lc
+pairs (D_i,Diff_{D_i}(D-D_i)), where the different
includes the conductor with coefficient one. Removing that conductor
-contribution defines an effective boundary on $D$; together with $D$ it
-is an slc pair whose adjoint line bundle is $M|_D$. The divisible
+contribution defines an effective boundary on D; together with D it
+is an slc pair whose adjoint line bundle is M|_D. The divisible
residue identifications hold in codimension one, with even powers
-eliminating residue signs at double crossings, and extend by $S_2$.
+eliminating residue signs at double crossings, and extend by S_2.
Each normalised adjoint is nef and semiample by
Assumption~mmp:lower-models and Lemma~mmp:comparison.
@@ -774,48 +767,48 @@ perturbation could change along the program. The following estimate
and line-bundle descent give that uniformity.
-Let $(Z,Delta)$ be a projective $Q$-factorial dlt rational pair of
-dimension $n$ with $L=K_Z+Delta$ nef. Put $P=Delta$ if the pair is
-klt, and $P=lfloorDeltarfloor$ otherwise. Fix $m>0$ with $mL$
+Let (Z,Delta) be a projective Q-factorial dlt rational pair of
+dimension n with L=K_Z+Delta nef. Put P=Delta if the pair is
+klt, and P=lfloorDeltarfloor otherwise. Fix m>0 with mL
Cartier. For every rational
[
0<ε<(1)/(4nm+2),
]
-every step of an LMMP for $L-ε P$ is $L$-trivial. On every
-model of the program, the transform of $L$ is nef, and its multiple by
-the same integer $m$ is Cartier.
+every step of an LMMP for L-ε P is L-trivial. On every
+model of the program, the transform of L is nef, and its multiple by
+the same integer m is Cartier.
Proof.
-Both $Delta-ε P$ and $Delta-tfrac12P$ are effective klt
-boundaries. If $R$ is a ray negative for $L-ε P$, nefness of $L$
-makes $R$ negative for $L-tfrac12P$. Choose a rational curve $C$
-spanning $R$ and satisfying the length bound
+Both Delta-ε P and Delta-tfrac12P are effective klt
+boundaries. If R is a ray negative for L-ε P, nefness of L
+makes R negative for L-tfrac12P. Choose a rational curve C
+spanning R and satisfying the length bound
[
-(L-tfrac12P)cdot Cle2n.
]
-Writing $a=Lcdot Cge0$ and $p=Pcdot C$, we have
+Writing a=Lcdot Cge0 and p=Pcdot C, we have
[
- ple2a+4n,qquad a<ε p,qquad (1-2ε)a<4nε.
+ ple2a+4n, a<ε p, (1-2ε)a<4nε.
]
-The choice of $ε$ gives $a<1/m$. Since $ma$ is a nonnegative
-integer, $a=0$.
+The choice of ε gives a<1/m. Since ma is a nonnegative
+integer, a=0.
-For the driving klt contraction $c:Zto B$, the line bundle $OO_Z(mL)$
-is numerically trivial over $B$. The line-bundle clause of the
-contraction theorem gives its descent to a line bundle $cA_B$ on $B$
-with no change of $m$ cite[Theorem~3.74(3)]{Fujino2017}. (One may also
+For the driving klt contraction c:Zto B, the line bundle OO_Z(mL)
+is numerically trivial over B. The line-bundle clause of the
+contraction theorem gives its descent to a line bundle cA_B on B
+with no change of m cite[Theorem~3.74(3)]{Fujino2017}. (One may also
see the unchanged index directly from relative basepoint freeness: two
consecutive sufficiently large powers descend, so their quotient
-descends.) The line bundle $cA_B$ is nef because its pullback
-$OO_Z(mL)$ is nef. For a flip $c^+:Z^+to B$, its pullback
-$(c^+)^*cA_B$ is the line bundle of the transformed $mL$. This proves,
-separately, that the transformed $L$ is nef and that $mL$ remains
-Cartier with the same $m$.
+descends.) The line bundle cA_B is nef because its pullback
+OO_Z(mL) is nef. For a flip c^+:Z^+to B, its pullback
+(c^+)^*cA_B is the line bundle of the transformed mL. This proves,
+separately, that the transformed L is nef and that mL remains
+Cartier with the same m.
The driving boundary remains klt throughout its LMMP. The transformed
-$Delta-tfrac12P$ remains effective and is smaller than the driving
-boundary, so it too is klt. The same estimate and the same integer $m$
+Delta-tfrac12P remains effective and is smaller than the driving
+boundary, so it too is klt. The same estimate and the same integer m
apply inductively at every subsequent step.
@@ -830,17 +823,17 @@ zero-Iitaka case and its klt perturbations.
[Inductive reduction to smooth nonvanishing]
Assume Assumption~mmp:lower-models. Assume also that every smooth
-projective $n$-fold with pseudo-effective canonical divisor has
-nonnegative Kodaira dimension. Then every projective lc $n$-fold with
+projective n-fold with pseudo-effective canonical divisor has
+nonnegative Kodaira dimension. Then every projective lc n-fold with
real boundary and pseudo-effective adjoint has a good log minimal
-model. In particular, every nef rational lc adjoint in dimension $n$
+model. In particular, every nef rational lc adjoint in dimension n
is semiample.
Proof.
Hashizume's reduction cite[Theorem~1.4]{Hashizume2018} gives
nonvanishing and log minimal models for projective lc pairs with real
-boundary in dimensions at most $n$. Pass to a $Q$-factorial dlt log
+boundary in dimensions at most n. Pass to a Q-factorial dlt log
minimal model. Fix the support of its boundary, impose the rational
affine constraints that its coefficient-one components remain equal
to one, and take a sufficiently small rational polytope around the
@@ -859,20 +852,20 @@ linear equations expressing its divisor and principal divisors has
rational data, so a real solution with positive coefficients has a
rational one.)
-We therefore work with a rational dlt pair $(Z,Delta)$ and
-$L=K_Z+Delta$ nef. Real nonvanishing implies rational nonvanishing
+We therefore work with a rational dlt pair (Z,Delta) and
+L=K_Z+Delta nef. Real nonvanishing implies rational nonvanishing
here: writing a real effective representative in terms of finitely
many divisors and principal divisors and solving the resulting rational
linear system with nonnegative rational coefficients produces a
-rational one. If $kappa(Z,L)ge1$, lower-dimensional good models give
+rational one. If kappa(Z,L)ge1, lower-dimensional good models give
a good minimal model by cite[Lemma~3.5]{FujinoGongyoRings2014}, and
-Lemma~mmp:comparison transfers semiampleness back to $Z$. It
-remains to treat $kappa(Z,L)=0$.
+Lemma~mmp:comparison transfers semiampleness back to Z. It
+remains to treat kappa(Z,L)=0.
The non-pseudo-effective perturbation.:
-Put $P=Delta$ in the klt case and $P=lfloorDeltarfloor$ otherwise.
-Suppose $L-ε P$ is not pseudo-effective for every sufficiently
-small $ε>0$. Choose rational $ε$ as in
+Put P=Delta in the klt case and P=lfloorDeltarfloor otherwise.
+Suppose L-ε P is not pseudo-effective for every sufficiently
+small ε>0. Choose rational ε as in
Lemma~mmp:trivial-perturbation, and run the klt LMMP with
scaling to a Mori fibre space
[
@@ -881,35 +874,35 @@ scaling to a Mori fibre space
]
Existence and termination in the non-pseudo-effective case are the
established klt results of [cite]. The program is crepant for
-$L$. The line-bundle descent in Lemma~mmp:trivial-perturbation,
+L. The line-bundle descent in Lemma~mmp:trivial-perturbation,
applied also to the final contraction, gives a nef rational divisor
-$A$ on $T$ with
+A on T with
[
K_{Z'}+Delta'sim_{Q}f^*A.
]
-The base $T$ has dimension less than $n$.
+The base T has dimension less than n.
-If $(Z,Delta)$ is klt, the transformed original pair $(Z',Delta')$
+If (Z,Delta) is klt, the transformed original pair (Z',Delta')
is klt as well. Ambro's descent theorem cite[Theorem~0.2]{Ambro2005}
-gives an effective rational boundary $Delta_T$ such that
-$(T,Delta_T)$ is klt and
+gives an effective rational boundary Delta_T such that
+(T,Delta_T) is klt and
[
Asim_{Q}K_T+Delta_T.
]
All the hypotheses hold: the total pair is globally klt and effective,
-$f$ is a projective contraction, and the adjoint is rationally
+f is a projective contraction, and the adjoint is rationally
linearly pulled back. In particular this use requires no conjecture on
semiampleness of a moduli divisor. Assumption~mmp:lower-models
-and Lemma~mmp:comparison make $A$ semiample.
-
-Otherwise $P'$ is effective and relatively ample, since
-$K_{Z'}+Delta'-ε P'$ is relatively antiample. Some component
-of the floor therefore dominates $T$. Take a crepant dlt blowup
-$b:(widetilde Z,widetildeDelta)to(Z',Delta')$ and choose the
-strict transform $S$ of such a component. The restriction
-$f_S=(fcirc b)|_S:Sto T$ is surjective. Crepancy and adjunction
-give an lc pair on $S$ whose nef adjoint is rationally linearly
-equivalent to $f_S^*A$; it is semiample by the lower-dimensional
+and Lemma~mmp:comparison make A semiample.
+
+Otherwise P' is effective and relatively ample, since
+K_{Z'}+Delta'-ε P' is relatively antiample. Some component
+of the floor therefore dominates T. Take a crepant dlt blowup
+b:(widetilde Z,widetildeDelta)to(Z',Delta') and choose the
+strict transform S of such a component. The restriction
+f_S=(fcirc b)|_S:Sto T is surjective. Crepancy and adjunction
+give an lc pair on S whose nef adjoint is rationally linearly
+equivalent to f_S^*A; it is semiample by the lower-dimensional
hypothesis. Semiampleness of a rational line bundle descends along a
proper surjection: use the equality of sections after Stein
factorisation for the connected fibre, and norms for the remaining
@@ -918,105 +911,105 @@ multiple nonzero simultaneously at all points of the chosen finite
fibre; such a section exists because each vanishing condition is a
proper linear subspace, and finitely many proper linear subspaces do
not cover a complex vector space. Its norm is nonzero at the
-downstairs point. Hence $A$ is semiample in this case as well, and
-crepancy transfers the conclusion back to $L$.
+downstairs point. Hence A is semiample in this case as well, and
+crepancy transfers the conclusion back to L.
The klt pseudo-effective perturbation.:
-We next settle the klt case where $L-εDelta$ is pseudo-effective
-for some small rational $ε>0$. Nonvanishing gives
+We next settle the klt case where L-εDelta is pseudo-effective
+for some small rational ε>0. Nonvanishing gives
[
- Lsim_{Q}G_0:=H_0+εDelta,qquad H_0ge0.
+ Lsim_{Q}G_0:=H_0+εDelta, H_0ge0.
]
-Take a resolution $p:Wto Z$ with reduced SNC divisor $D$ containing
-all exceptional divisors and the strict support of $H_0+Delta$. Then
+Take a resolution p:Wto Z with reduced SNC divisor D containing
+all exceptional divisors and the strict support of H_0+Delta. Then
[
- K_W+D=p^*L+R,qquad Rge0.
+ K_W+D=p^*L+R, Rge0.
]
-Every component of $D$ has positive coefficient in
-$G_W:=p^*G_0+Rsim_{Q}K_W+D$. Indeed, at strict boundary components
+Every component of D has positive coefficient in
+G_W:=p^*G_0+Rsim_{Q}K_W+D. Indeed, at strict boundary components
the difference between coefficient one and a klt boundary coefficient
is positive, and at exceptional components the coefficient contributed
by the adjoint comparison is the positive log discrepancy. The
-pushforward $p_*G_W$ is bounded coefficientwise by a fixed multiple of
-$G_0$, and sections inject under birational pushforward, so
+pushforward p_*G_W is bounded coefficientwise by a fixed multiple of
+G_0, and sections inject under birational pushforward, so
[
0lekappa(W,K_W+D)lekappa(Z,G_0)=0.
]
-Take a $Q$-factorial dlt log minimal model $(V,D_V)$ of $(W,D)$, and
+Take a Q-factorial dlt log minimal model (V,D_V) of (W,D), and
fix a common resolution
[
Wxleftarrow{q}Uxrightarrow{v}V.
]
-Its boundary is reduced. On $U$ the comparison divisor of
-Lemma~mmp:comparison is exceptional over $V$; pushing $q^*G_W$
-forward by $v$ gives
+Its boundary is reduced. On U the comparison divisor of
+Lemma~mmp:comparison is exceptional over V; pushing q^*G_W
+forward by v gives
[
- K_V+D_Vsim_{Q}G_V=sum_ib_iD_i,qquad b_i>0,
+ K_V+D_Vsim_{Q}G_V=sum_ib_iD_i, b_i>0,
qquadSupp G_V=D_V.
]
-(For completeness: at a component extracted on $V$, positivity also
-holds since its log discrepancy over $(W,D)$ is zero, its centre lies
-in $D$, and the pullback of the full-support effective $G_W$ has
-positive order there. At a prime retained on $V$, there is no
+(For completeness: at a component extracted on V, positivity also
+holds since its log discrepancy over (W,D) is zero, its centre lies
+in D, and the pullback of the full-support effective G_W has
+positive order there. At a prime retained on V, there is no
comparison-divisor coefficient.) Effectivity and exceptionality in the
comparison preserve the adjoint section ring, so the nef adjoint on
-$V$ still has Iitaka dimension zero.
+V still has Iitaka dimension zero.
The small-perturbation hypothesis of Theorem~thm:signed is now
-explicit. If $D_Vne0$, choose rational $c$ with $0<c<min_ib_i$; then
+explicit. If D_Vne0, choose rational c with 0<c<min_ib_i; then
[
K_V+(1-c)D_Vsim_{Q}sum_i(b_i-c)D_ige0.
]
-If $D_V=0$, pseudo-effectivity is immediate.
+If D_V=0, pseudo-effectivity is immediate.
Proposition~prop:boundary-restriction supplies the required
-semiampleness on $D_V$. Theorem~thm:signed then makes $K_V+D_V$
+semiampleness on D_V. Theorem~thm:signed then makes K_V+D_V
abundant. Its Iitaka dimension is zero, so its numerical dimension is
-zero. For an ample divisor $H_V$ on $V$ we get
+zero. For an ample divisor H_V on V we get
[
G_Vcdot H_V^{n-1}=(K_V+D_V)cdot H_V^{n-1}=0.
]
-Effectivity forces $G_V=0$, and its full support then gives $D_V=0$.
+Effectivity forces G_V=0, and its full support then gives D_V=0.
-Since $G_W=p^*G_0+R$ with $Rge0$, on the same common resolution
+Since G_W=p^*G_0+R with Rge0, on the same common resolution
[
- 0le q^*p^*G_0le q^*G_W,qquad v_*q^*G_W=G_V=0.
+ 0le q^*p^*G_0le q^*G_W, v_*q^*G_W=G_V=0.
]
-Thus $q^*p^*G_0$ is effective and $v$-exceptional. It is nef because
-it is rationally linearly equivalent to $(pcirc q)^*L$. The
-negativity lemma for $v$ forces it to vanish. Hence $G_0=0$ and
-$Lsim_{Q}0$. Together with the preceding cases this settles the klt
-good-model assertion in dimension $n$: for a non-nef pseudo-effective
+Thus q^*p^*G_0 is effective and v-exceptional. It is nef because
+it is rationally linearly equivalent to (pcirc q)^*L. The
+negativity lemma for v forces it to vanish. Hence G_0=0 and
+Lsim_{Q}0. Together with the preceding cases this settles the klt
+good-model assertion in dimension n: for a non-nef pseudo-effective
klt adjoint, first take its klt log minimal model and apply what has
just been proved.
The remaining non-klt case.:
Finally suppose the original pair is not klt and a small floor
-perturbation $L-delta P$ is pseudo-effective. Choose rational
-$ε>0$ with $2εledelta$, small enough to keep the
+perturbation L-delta P is pseudo-effective. Choose rational
+ε>0 with 2εledelta, small enough to keep the
perturbed boundaries effective and klt. Convexity of the
-pseudo-effective cone, applied between $L$ and $L-delta P$, makes
+pseudo-effective cone, applied between L and L-delta P, makes
both
[
- L-ε P,qquad L-2ε P,qquad P=lfloorDeltarfloor,
+ L-ε P, L-2ε P, P=lfloorDeltarfloor,
]
pseudo-effective klt adjoints. Their Iitaka dimensions are zero:
nonvanishing gives the lower bound, and adding the effective
-perturbation bounds each by $kappa(Z,L)=0$. The preceding klt case
+perturbation bounds each by kappa(Z,L)=0. The preceding klt case
gives good models for both of these perturbed adjoints, so their
Nakayama numerical dimensions are zero. Since
[
2(L-ε P)=L+(L-2ε P),
]
and the last summand has an effective rational representative,
-monotonicity and homogeneity of $kappa_sigma$ give
+monotonicity and homogeneity of kappa_sigma give
[
kappa_sigma(Z,L)lekappa_sigma(Z,2(L-ε P))=0.
]
-The nef divisor $L$ is therefore numerically trivial. Nonvanishing is
+The nef divisor L is therefore numerically trivial. Nonvanishing is
now already available in this finishing case: write
-$Lsim_{Q}Ege0$. For an ample divisor $H$, $Ecdot H^{n-1}=0$ forces
-$E=0$. Hence $Lsim_{Q}0$, without any further nonvanishing or
+Lsim_{Q}Ege0. For an ample divisor H, Ecdot H^{n-1}=0 forces
+E=0. Hence Lsim_{Q}0, without any further nonvanishing or
abundance premise.
This settles the remaining zero-Iitaka case, so every rational nef
@@ -1032,136 +1025,136 @@ nef rational lc pair.
%======================================================================
The induction, at its key step, has a nef log canonical divisor
-$L=K_X+D$ on a $Q$-factorial dlt pair $(X,D)$ with $D$ reduced. The
+L=K_X+D on a Q-factorial dlt pair (X,D) with D reduced. The
hypothesis of the signed criterion is that some rational-line
-representative of $L$ --- not necessarily the boundary itself --- is
-supported on $D$ with either signs. The difficulty is that $L|_D$ is
-semiample but $L$ itself is only nef.
+representative of L --- not necessarily the boundary itself --- is
+supported on D with either signs. The difficulty is that L|_D is
+semiample but L itself is only nef.
The engine of this section converts that hypothesis into abundance.
-The outline, already given on page~pageref{par:engineering}, is
+The outline, already given on page~pageref{:engineering}, is
repeated here in numerical form so the reader knows what the
individual lemmas are for.
-The semiample $L|_D$ gives a morphism
+The semiample L|_D gives a morphism
[
- phi:D→ P:=PP^s,qquad
+ phi:D→ P:=PP^s,
N_0|_Dsimeqphi^*OO_P(1)
]
-for a sufficiently divisible positive integer $m$, where $N_0:=OO_X(mL)$.
-Write $s_0$ for the rational section of $N_0$ having divisor $mG$, where
-$G=G_+-G_-$ comes from the signed representative and $G_pm,Ege0$. Let
-$r+1$ be the dimension of the image of a positive-coefficient
-component of $D$ in $P$; a lemma below shows that every other
-component has image of dimension at most $r$, and that $G_+$ restricted
-to $Supp G_-cup D_0$ has image of dimension less than $r$.
+for a sufficiently divisible positive integer m, where N_0:=OO_X(mL).
+Write s_0 for the rational section of N_0 having divisor mG, where
+G=G_+-G_- comes from the signed representative and G_pm,Ege0. Let
+r+1 be the dimension of the image of a positive-coefficient
+component of D in P; a lemma below shows that every other
+component has image of dimension at most r, and that G_+ restricted
+to Supp G_-cup D_0 has image of dimension less than r.
The root construction of
Proposition~prop:signed-construction produces, from this data,
-a projective variety $mathcal X$calX$ with a reduced Cartier divisor
-$Ssubsetmathcal X$calX$ and a finite representable morphism
-$Sto Dtimes_Pmathcal P$calP$ whose image covers the positive boundary,
-such that $S$ is (generically) its own root line and carries a
+a projective variety mathcal XcalX with a reduced Cartier divisor
+Ssubsetmathcal XcalX and a finite representable morphism
+Sto Dtimes_Pmathcal PcalP whose image covers the positive boundary,
+such that S is (generically) its own root line and carries a
Stein factorisation
[
Sxrightarrow{g}mathcal YcalYxrightarrow{p}PP^s
]
-with $g$ finite representable and $p$ a projective bundle. The
+with g finite representable and p a projective bundle. The
factorisation is the geometric output that the Hodge step of
Section~sec:hodge consumes. That step lifts functions on the
positive boundary to every infinitesimal thickening, deformes them off
-$D$, and produces a fixed moveable family of subvarieties on which $L$
+D, and produces a fixed moveable family of subvarieties on which L
is trivial. Descent along the resulting nef reduction
(Section~sg:descent) then proves abundance.
The full theorem is:
[Signed-boundary abundance criterion]
-Let $X$ be a projective $Q$-factorial dlt variety of dimension $n$,
-let $D=sum_iD_i$ be a reduced divisor whose support is the
+Let X be a projective Q-factorial dlt variety of dimension n,
+let D=sum_iD_i be a reduced divisor whose support is the
coefficient-one part of the boundary, and let
-$L=K_X+D$. Suppose:
+L=K_X+D. Suppose:
[label=(roman*)]
-item $L$ is nef;
-item $L-cD$ is pseudo-effective for some rational $c>0$;
-item $L|_D$ is semiample as a rational line bundle on the reduced
-scheme $D$;
-item the support $D$ is nonempty and $0<nu(L)<n$.
-
-Then $kappa(L)=nu(L)$.
-
-If instead $nu(L)=n$, then $L$ is big (the numerical criterion for
-a nef divisor). If $nu(L)=0$, the case is either $D=0$, in which
-case $Lsim_{Q}0$ and there is nothing to prove, or the argument
-below applies after shrinking $D$ to
+ L is nef;
+ L-cD is pseudo-effective for some rational c>0;
+ L|_D is semiample as a rational line bundle on the reduced
+scheme D;
+ the support D is nonempty and 0<nu(L)<n.
+
+Then kappa(L)=nu(L).
+
+If instead nu(L)=n, then L is big (the numerical criterion for
+a nef divisor). If nu(L)=0, the case is either D=0, in which
+case Lsim_{Q}0 and there is nothing to prove, or the argument
+below applies after shrinking D to
Proposition~prop:inductive-reduction.
The proof occupies the remainder of this section and the next; it does
not use the induction hypothesis, only established results of the
-minimal model program. We write $Lsim_{Q}sum_ia_iD_i$ for the
-signed representative, allowing $a_iinQ$ of any sign, and decompose
-$G=G_+-G_-$ into effective parts with disjoint prime supports;
-$D_0=sum_{a_i=0}D_i$ collects the coefficient-zero components.
+minimal model program. We write Lsim_{Q}sum_ia_iD_i for the
+signed representative, allowing a_iinQ of any sign, and decompose
+G=G_+-G_- into effective parts with disjoint prime supports;
+D_0=sum_{a_i=0}D_i collects the coefficient-zero components.
[Numerical location of the boundary]
-In the setting of Theorem~thm:signed, assume $0<nu(L)<n$.
-Put $r=nu(L)-1$. Then:
+In the setting of Theorem~thm:signed, assume 0<nu(L)<n.
+Put r=nu(L)-1. Then:
[label=(roman*)]
-item every component of $D$ has $phi$-image of dimension at most $r$;
-item some component of $G_+$ has $phi$-image of dimension exactly $r$;
-item the locus $Supp G_+cap(Supp G_-cup D_0)$ has $phi$-image of
- dimension less than $r$ (empty when $r=0$).
+ every component of D has phi-image of dimension at most r;
+ some component of G_+ has phi-image of dimension exactly r;
+ the locus Supp G_+cap(Supp G_-cup D_0) has phi-image of
+ dimension less than r (empty when r=0).
Proof.
-Part (i). Intersect the pseudo-effective class $L-cD$ with the product
-of nef classes $L^vH^{n-v-1}$, $v=nu(L)$, and $H$ ample. Every term
-is nonnegative: $L^vH^{n-v-1}$ is nef intersection of $L$, and $(L-cD)$
+Part (i). Intersect the pseudo-effective class L-cD with the product
+of nef classes L^vH^{n-v-1}, v=nu(L), and H ample. Every term
+is nonnegative: L^vH^{n-v-1} is nef intersection of L, and (L-cD)
is pseudo-effective. The result is
[
0le(L-cD)L^vH^{n-v-1}
=-csum_iD_iL^vH^{n-v-1}.
]
-Each $D_iL^vH^{n-v-1}$ is nonnegative, so all vanish. If $v=0$,
-ampleness forces $D=0$, contradicting the hypothesis $Dne0$. For
-$v>0$, semiampleness of $L|_D$ identifies the numerical dimension of
-each $D_i$'s image in $P$ with the numerical dimension of $L|_{D_i}$,
-which equals the intersection $D_iL^{nu(L)-1}H^{n-nu(L)}=dimphi(D_i)$.
+Each D_iL^vH^{n-v-1} is nonnegative, so all vanish. If v=0,
+ampleness forces D=0, contradicting the hypothesis Dne0. For
+v>0, semiampleness of L|_D identifies the numerical dimension of
+each D_i's image in P with the numerical dimension of L|_{D_i},
+which equals the intersection D_iL^{nu(L)-1}H^{n-nu(L)}=dimphi(D_i).
This intersection is zero by the vanishing just proved, so the image
-dimension of every $D_i$ is at most $r=nu(L)-1$.
+dimension of every D_i is at most r=nu(L)-1.
Part (ii). By the same vanishing,
[
0<L^{nu(L)}H^{n-nu(L)}=sum_ia_iD_iL^rH^{n-v},
]
-so at least one $a_i>0$ and at least one term is strictly positive.
-The corresponding component of $G_+$ attains image dimension exactly
-$r$.
+so at least one a_i>0 and at least one term is strictly positive.
+The corresponding component of G_+ attains image dimension exactly
+r.
-Part (iii). Consider the symmetric matrix $Q_{ij}=D_iD_jL^rH^{n-r-2}$.
-Distinct effective $Q$-Cartier divisors have effective intersection
+Part (iii). Consider the symmetric matrix Q_{ij}=D_iD_jL^rH^{n-r-2}.
+Distinct effective Q-Cartier divisors have effective intersection
cycles, so the off-diagonal entries are nonnegative. The mixed Hodge
index theorem gives the ambient intersection form at most one positive
direction: approximate by ample classes and apply the ordinary Hodge
-index theorem on complete-intersection surfaces. The class $L$ is
-isotropic and orthogonal to every $D_i$ (this is the vanishing from
-part (i)), while its pairing with $H$ is strictly positive:
+index theorem on complete-intersection surfaces. The class L is
+isotropic and orthogonal to every D_i (this is the vanishing from
+part (i)), while its pairing with H is strictly positive:
[
- Q(L,L)=L^{r+2}H^{n-r-2}=0,quad
- Q(L,D_i)=D_iL^{r+1}H^{n-r-2}=0,quad
+ Q(L,L)=L^{r+2}H^{n-r-2}=0,
+ Q(L,D_i)=D_iL^{r+1}H^{n-r-2}=0,
Q(L,H)=L^{r+1}H^{n-r-1}>0.
]
-The intersection form on the orthogonal complement of $L$ is therefore
-negative semidefinite, and this holds on the span of the $D_i$. Pulling
-back to a resolution preserves the conclusion, so smoothness of $X$
+The intersection form on the orthogonal complement of L is therefore
+negative semidefinite, and this holds on the span of the D_i. Pulling
+back to a resolution preserves the conclusion, so smoothness of X
is not needed. But any strictly positive off-diagonal entry would
contribute a positive direction to the span; the entries therefore
-have image dimension strictly less than $r$ wherever their images have
-$r$-dimensional support, forcing the locus
-$Supp G_+ cap(Supp G_-cup D_0)$ to have $phi$-image of dimension
-less than $r$.
+have image dimension strictly less than r wherever their images have
+r-dimensional support, forcing the locus
+Supp G_+ cap(Supp G_-cup D_0) to have phi-image of dimension
+less than r.
We will also place the Stein image in a smooth target: this allows
@@ -1170,71 +1163,67 @@ used on ordinary scheme charts. The infinitesimal neighborhoods will
later be cut down to a projective fiber before they are algebraized.
-Choose $ell$ divisible by $m$ and every nonzero integer $|ma_i|$,
-and set $a=ell/m$. There is a normal tame Deligne--Mumford stack
-$mathcal X$,calX$, considered on a neighborhood of a reduced Cartier
-divisor $S$ of pure dimension $N=n-1$, together with a morphism
-to $X$ and a reduced boundary
-$T$ having no component in common with $S$, with the following
+Choose ell divisible by m and every nonzero integer |ma_i|,
+and set a=ell/m. There is a normal tame Deligne--Mumford stack
+mathcal X,calX, considered on a neighborhood of a reduced Cartier
+divisor S of pure dimension N=n-1, together with a morphism
+to X and a reduced boundary
+T having no component in common with S, with the following
properties.
[label=(roman*)]
-item
-The pair $(mathcal X,S+T)$calX,S+T)$ is log canonical, and a fixed
+
+The pair (mathcal X,S+T)calX,S+T) is log canonical, and a fixed
isomorphism of reflexive sheaves is given by
[
omega_{mathcal X}calX(S+T)simeqOO_{mathcal X}calX(aS).
tag{(ast_{adj})}
]
-The support of $T$ is locally set-theoretically principal.
-The ambient space and $S$ are Cohen--Macaulay on scheme charts.
-Off $T$, the ambient space is Gorenstein and the displayed
+The support of T is locally set-theoretically principal.
+The ambient space and S are Cohen--Macaulay on scheme charts.
+Off T, the ambient space is Gorenstein and the displayed
isomorphism is ordinary Cartier adjunction data.
-item
-Put $J=(Scap T)_{red}$. Both $S$ and $J$ are Du Bois
-on scheme charts, the ideal $mathcal I_calI_{J/S}$ is maximal
+Put J=(Scap T)_{red}. Both S and J are Du Bois
+on scheme charts, the ideal mathcal I_calI_{J/S} is maximal
Cohen--Macaulay, and
[
mathcal B:=calB:=
mathcal Hom_S(calHom_S(mathcal I_calI_{J/S},omega_S)
simeqOO_S(aS).
]
-The identification agrees off $J$ with the residue convention
+The identification agrees off J with the residue convention
specified by (sg:ambient-adjunction).
-item
There is a finite representable morphism
[
S→ Dtimes_Pmathcal PcalP
]
-whose image covers $Supp G_+$. The line $OO_S(S)$ is the pullback
-of $C$. The image of $J$ in $P$ has dimension less than $r$,
-whereas the image of $S$ has dimension $r$.
-
-item
-There is a smooth projective bundle $p:mathcal YcalYtomathcal P$calP$
-and a representable projective morphism $g:Stomathcal Y$.calY$.
-This is a morphism from the reduced divisor $S$; no extension
-to a neighborhood of $S$ in $mathcal X$calX$ is asserted here.
-Writing $h=pcirc g$ and
-$mathcal T=calT=Spec_{mathcal P}calPh_*OO_S$, the morphism $g$ factors
-through a closed embedding $mathcal T↪calT↪mathcal Y$,calY$,
+whose image covers Supp G_+. The line OO_S(S) is the pullback
+of C. The image of J in P has dimension less than r,
+whereas the image of S has dimension r.
+
+There is a smooth projective bundle p:mathcal YcalYtomathcal PcalP
+and a representable projective morphism g:Stomathcal Y.calY.
+This is a morphism from the reduced divisor S; no extension
+to a neighborhood of S in mathcal XcalX is asserted here.
+Writing h=pcirc g and
+mathcal T=calT=Spec_{mathcal P}calPh_*OO_S, the morphism g factors
+through a closed embedding mathcal T↪calT↪mathcal Y,calY,
and
[
- g_*OO_S=OO_{mathcal T}calT,qquad
- OO_S(S)=g^*C,qquad mathcal B=g^*C^a.calB=g^*C^a.
+ g_*OO_S=OO_{mathcal T}calT,
+ OO_S(S)=g^*C, mathcal B=g^*C^a.calB=g^*C^a.
]
-item
For a separated quasi-projective scheme chart
-$Utomathcal Y$calY$ that is etale and of finite type, put
-$S_U=Stimes_{mathcal Y}calYU$.
-The etale morphism $S_Uto S$ extends compatibly to every
-nilpotent thickening $qS$ in $mathcal X$.calX$.
-Only this etale chart is extended; an extension of $g$ to $qS$
+Utomathcal YcalY that is etale and of finite type, put
+S_U=Stimes_{mathcal Y}calYU.
+The etale morphism S_Uto S extends compatibly to every
+nilpotent thickening qS in mathcal X.calX.
+Only this etale chart is extended; an extension of g to qS
is not needed to define the chart.
-The resulting $qS_U$ are quasi-projective schemes, separated
-and quasi-finite over $X$.
+The resulting qS_U are quasi-projective schemes, separated
+and quasi-finite over X.
All the adjunction and duality identifications are compatible
with changes of etale chart.
@@ -1247,19 +1236,19 @@ verifies that the infinitesimal neighborhoods have the scheme
charts required for formal lifting.
smallskipnoindentRoots and a fixed adjunction isomorphism.
-Let $rho:mathcal RcalRto X$ be the normalized simultaneous root stack of the Cartier
-divisors $mG_+$, $mG_-$, and $mD_0$, taking an $ell$-th root of
+Let rho:mathcal RcalRto X be the normalized simultaneous root stack of the Cartier
+divisors mG_+, mG_-, and mD_0, taking an ell-th root of
each divisor together with its section; the construction allows
empty divisors. Denote the tautological root divisors by
-$S_+,S_-,S_0$ on $mathcal R$.calR$. Their reducedness can be checked
-at a generic prime of downstairs multiplicity $b$. The chosen
-divisibility conditions give $bmidell$. A normalized chart of
-$y^ell=x^b$ has ramification index $ell/b$, and $y$ has order
-one. Since $m$ divides $ell$, this also applies to $D_0$.
+S_+,S_-,S_0 on mathcal R.calR. Their reducedness can be checked
+at a generic prime of downstairs multiplicity b. The chosen
+divisibility conditions give bmidell. A normalized chart of
+y^ell=x^b has ramification index ell/b, and y has order
+one. Since m divides ell, this also applies to D_0.
The divisors are Cartier, so normality and generic reducedness
give their reducedness everywhere.
-Put $B=S_++S_-+S_0$ on $mathcal R$.calR$. With this full reduced
+Put B=S_++S_-+S_0 on mathcal R.calR. With this full reduced
boundary, log ramification gives a log
canonical pair and the relation
[
@@ -1274,29 +1263,29 @@ between the two sides is a torsion integral Weil class, represented by
(omega_{mathcal R}calR(B)
⊗ OO_{mathcal R}calR(-a(S_+-S_-)))^{**}.
]
-Choose a periodicity isomorphism $mathcal F^calF^{[q]}simeqOO_{mathcal R}calR$,
+Choose a periodicity isomorphism mathcal F^calF^{[q]}simeqOO_{mathcal R}calR,
where square brackets denote reflexive powers. It defines multiplication
in the finite algebra
[
mathcal A=⊕_calA=⊕_{i=0}^{q-1}mathcal F^calF^{[i]}.
]
-Let $tau:mathcal R^calR^daggertomathcal R$calR$ be the normalization
-of $Spec_{mathcal R}calRmathcal A$.calA$. In codimension one this is
+Let tau:mathcal R^calR^daggertomathcal RcalR be the normalization
+of Spec_{mathcal R}calRmathcal A.calA. In codimension one this is
the ordinary cover of a torsion line bundle and is etale. The
cover preserves log canonicity, klt ambient singularities, and
-the reduced Cartier boundary. From now on $S_+,S_-,S_0$ denote
-their pullbacks to $mathcal R^calR^dagger$.
+the reduced Cartier boundary. From now on S_+,S_-,S_0 denote
+their pullbacks to mathcal R^calR^dagger.
Tautological evaluation on this cover trivializes the pullback
-of $mathcal F$calF$ in codimension one and hence fixes the adjoint
+of mathcal FcalF in codimension one and hence fixes the adjoint
isomorphism. Because this evaluation is a sheaf morphism
on the cover stack itself, the adjoint isomorphism is equivariant
on every atlas. This equivariance will be needed for the
quotient below.
smallskipnoindentSeparating the positive and negative parts.
-Let $p_1:mathcal RtildecalRtildetomathcal R^calR^dagger$ be the
-normalized blowup of the ideal of $S_+cap S_-$. The spaces
+Let p_1:mathcal RtildecalRtildetomathcal R^calR^dagger be the
+normalized blowup of the ideal of S_+cap S_-. The spaces
constructed so far are related by
[
mathcal RtildecalRtildexrightarrow{p_1}mathcal R^calR^dagger
@@ -1312,33 +1301,33 @@ stratum, generically SNC. Separate normalized Kummer charts
and purity for the index cover at the smooth generic locus
give the same description upstairs.
-The tautological exceptional divisor $E$ is consequently reduced
+The tautological exceptional divisor E is consequently reduced
Cartier, and the divisors
[
- S'_+=p_1^*S_+-E,qquad S'_-=p_1^*S_--E
+ S'_+=p_1^*S_+-E, S'_-=p_1^*S_--E
]
-are reduced Cartier and disjoint on $mathcal Rtilde$.calRtilde$.
+are reduced Cartier and disjoint on mathcal Rtilde.calRtilde.
Since no codimension-two two-branch stratum is contained
-in $S_0$, its pullback $p_1^*S_0$ is reduced Cartier and has no
+in S_0, its pullback p_1^*S_0 is reduced Cartier and has no
exceptional component. The total boundary
[
S'_++S'_-+E+p_1^*S_0
]
is crepant and log canonical. To check the ambient singularities
-as well, denote this boundary on a chart $V'$ by $B'$. The
+as well, denote this boundary on a chart V' by B'. The
crepant equality gives log canonicity for every valuation.
-Outside $B'$, the blowup is an isomorphism to the old klt
-boundary complement. A divisor $F$ with
-$a(F;V',B')=0$ therefore has center in $Supp B'$. Since $B'$
-is effective Cartier, $ord_F(B')>0$, whence
+Outside B', the blowup is an isomorphism to the old klt
+boundary complement. A divisor F with
+a(F;V',B')=0 therefore has center in Supp B'. Since B'
+is effective Cartier, ord_F(B')>0, whence
[
a(F;V',0)=a(F;V',B')+ord_F(B')>0.
]
Positive pair discrepancies also remain positive after the
-boundary is dropped. Thus $V'$ is klt even at its
-higher-codimension singular loci. On $mathcal Rtilde$,calRtilde$,
-put $S=S'_+$ and $T=S'_-+E+p_1^*S_0$. Near $S$, the section of the disjoint
-divisor $S'_-$ is a unit, and the fixed linear equivalence reads
+boundary is dropped. Thus V' is klt even at its
+higher-codimension singular loci. On mathcal Rtilde,calRtilde,
+put S=S'_+ and T=S'_-+E+p_1^*S_0. Near S, the section of the disjoint
+divisor S'_- is a unit, and the fixed linear equivalence reads
[
omega(S+T)simeqOO(aS).
]
@@ -1346,27 +1335,27 @@ Klt singularities make the ambient charts Cohen--Macaulay.
All the boundary divisors in this display are Cartier, so the
isomorphism also makes these charts Gorenstein.
-We also need finiteness of $Sto S_+$ on $mathcal R^calR^dagger$.
-Locally write $S_+=(u=0)$ and $S_-=(v=0)$. Before normalization,
-the strict transform of $S_+$ lies in the blowup chart with
-ratio $u/v$, where its equation is $u/v=0$. Thus each fiber of
-this strict transform over $S_+$ has at most one point.
-Properness and finite normalization prove finiteness of $Sto S_+$.
-A codimension-one point of $Scap T$ therefore lies over a
+We also need finiteness of Sto S_+ on mathcal R^calR^dagger.
+Locally write S_+=(u=0) and S_-=(v=0). Before normalization,
+the strict transform of S_+ lies in the blowup chart with
+ratio u/v, where its equation is u/v=0. Thus each fiber of
+this strict transform over S_+ has at most one point.
+Properness and finite normalization prove finiteness of Sto S_+.
+A codimension-one point of Scap T therefore lies over a
codimension-two intersection downstairs. The generic SNC
-description shows that $T|_S$ is generically reduced. Being
-Cartier on the Cohen--Macaulay scheme $S$, it is reduced
+description shows that T|_S is generically reduced. Being
+Cartier on the Cohen--Macaulay scheme S, it is reduced
everywhere.
smallskipnoindentRetaining the required root line.
The construction so far has supplied reduced Cartier divisors
and fixed adjunction data. We now remove the root characters
-that are unnecessary, while preserving the line attached to $S$.
-On $mathcal Rtilde$calRtilde$ the line $OO(S-S'_-)$ has $ell$-th power equal to the pullback
-of $N_0$ and hence defines a morphism to the gerbe of $ell$-th
-roots of $N_0$ on $X$. Denote this gerbe by $mathcal G$,calG$, and let
-$kappa:mathcal RtildecalRtildetomathcal X$calX$ be the relative
-coarse-space morphism over $mathcal G$.calG$. On a trivializing chart, we quotient
+that are unnecessary, while preserving the line attached to S.
+On mathcal RtildecalRtilde the line OO(S-S'_-) has ell-th power equal to the pullback
+of N_0 and hence defines a morphism to the gerbe of ell-th
+roots of N_0 on X. Denote this gerbe by mathcal G,calG, and let
+kappa:mathcal RtildecalRtildetomathcal XcalX be the relative
+coarse-space morphism over mathcal G.calG. On a trivializing chart, we quotient
by the kernel of the finite-group character on the indicated
root line. The quotients are ordinary quasi-projective schemes:
the covers used above are finite, the blowup is projective,
@@ -1374,37 +1363,37 @@ and finite quotients preserve quasi-projectivity. These
chartwise quotients patch.
Before taking the quotient, the section of the disjoint divisor
-$S'_-$ is a unit near $S$. There the retained line
-$OO(S-S'_-)$ identifies with $OO(S)$, so the line of $S$ and
+S'_- is a unit near S. There the retained line
+OO(S-S'_-) identifies with OO(S), so the line of S and
its section both descend through the kernel quotient. Continue
-to write $S,T$ for their reduced images under $kappa$; they
-now lie on $mathcal X$.calX$. In particular $S$
+to write S,T for their reduced images under kappa; they
+now lie on mathcal X.calX. In particular S
remains Cartier. The finite-group norm of a local equation
-for $T$ upstairs makes its image set-theoretically principal;
-Cartierness of the reduced image $T$ is not needed.
+for T upstairs makes its image set-theoretically principal;
+Cartierness of the reduced image T is not needed.
Divisorial ramification occurs only on the boundary. Log
ramification thus preserves log canonicity and identifies the
descended adjoint isomorphism with
-(sg:ambient-adjunction). The ambient chart and $S$ are
+(sg:ambient-adjunction). The ambient chart and S are
Cohen--Macaulay because their structure sheaves are invariant
direct summands of the finite CM covers.
More explicitly, the kernel acts trivially on the retained root
-line and hence on $OO(aS)$. Equivariance of the fixed
-isomorphism gives the same kernel character on $omega(S+T)$.
+line and hence on OO(aS). Equivariance of the fixed
+isomorphism gives the same kernel character on omega(S+T).
Taking invariants descends the isomorphism, and log ramification
-identifies the invariant log dualizing sheaf. Off $T$, this
+identifies the invariant log dualizing sheaf. Off T, this
isomorphism makes the canonical sheaf invertible, giving the
asserted Gorenstein property.
smallskipnoindentDuality on the boundary.
-The reduced supports $S$ and $J$ are unions of log canonical
+The reduced supports S and J are unions of log canonical
centers, since intersections of lc centers are again unions
of lc centers. Both are therefore Du Bois by
cite[Theorems~1.4 and~1.7]{KollarKovacs2010}.
To identify the ideal and its dual, work on a quotient chart
-and denote the finite cover above by $f:S^{up}to S$.
-Reducedness of $T^{up}|_{S^{up}}$ gives
+and denote the finite cover above by f:S^{up}to S.
+Reducedness of T^{up}|_{S^{up}} gives
[
mathcal I_calI_{J/S}
=
@@ -1427,64 +1416,64 @@ Apply finite-map duality with trace and then take invariants:
simeqOO_S(aS).
]
The last isomorphism comes from Cartier adjunction upstairs
-and descent of the line of $S$. Finite-map duality applies
+and descent of the line of S. Finite-map duality applies
in the presence of ramification; it requires no invertibility
-of $omega_S$ itself. Off $J$, the identification is the
+of omega_S itself. Off J, the identification is the
ordinary residue fixed by the ambient isomorphism. Normalized
trace preserves this residue convention, so the identifications
agree on overlaps.
smallskipnoindentThe target of the lifting argument.
-On $S$, the root gerbe of $N_0$ is $Dtimes_Pmathcal P$.calP$.
+On S, the root gerbe of N_0 is Dtimes_Pmathcal P.calP.
Finiteness of the strict divisor, proved above, and removal
of the relative inertia kernel give a finite representable
-map $Sto Dtimes_Pmathcal P$.calP$. Its image covers the positive
-boundary, and the root line is $OO_S(S)=C$.
+map Sto Dtimes_Pmathcal P.calP. Its image covers the positive
+boundary, and the root line is OO_S(S)=C.
Lemma~sg:numerical-boundary therefore gives image dimension
-$r$ for $S$ in $P$. To check the smaller image of $J$, recall
-that before the quotient $S'_-$ is disjoint from $S$, the
-exceptional divisor $E$ maps into $Supp G_+capSupp G_-$,
-and $p_1^*S_0$ maps into $D_0$. Taking the quotient changes none
-of these images in $X$. The image of $J$ in $D$ is consequently
+r for S in P. To check the smaller image of J, recall
+that before the quotient S'_- is disjoint from S, the
+exceptional divisor E maps into Supp G_+capSupp G_-,
+and p_1^*S_0 maps into D_0. Taking the quotient changes none
+of these images in X. The image of J in D is consequently
contained in
[
Supp G_+cap(Supp G_-cup D_0).
]
-The same lemma gives image dimension less than $r$ for this
-locus in $P$. We have thus obtained a
-representable projective morphism $h:Stomathcal P$;calP$; it remains
+The same lemma gives image dimension less than r for this
+locus in P. We have thus obtained a
+representable projective morphism h:Stomathcal P;calP; it remains
to embed its Stein image in the required smooth bundle.
The Stein algebra defines a finite stack
-$mathcal T=calT=Spec_{mathcal P}calPh_*OO_S$ over $mathcal P$.calP$.
+mathcal T=calT=Spec_{mathcal P}calPh_*OO_S over mathcal P.calP.
This coherent algebra is generated as a module by vector
-bundles on $mathcal P$.calP$. Indeed, decompose it by the finitely
-many inertia characters, twist each summand by a power of $C$
-so that it descends to $P$, and use Serre generation there.
-The resulting vector-bundle surjection embeds $mathcal T$calT$ in
-a vector bundle over $mathcal P$.calP$. Since $mathcal T$calT$ is
+bundles on mathcal P.calP. Indeed, decompose it by the finitely
+many inertia characters, twist each summand by a power of C
+so that it descends to P, and use Serre generation there.
+The resulting vector-bundle surjection embeds mathcal TcalT in
+a vector bundle over mathcal P.calP. Since mathcal TcalT is
proper over the base, it remains closed in the projective
-completion. This completion is $mathcal Y$calY$ and gives the map
-$g:Stomathcal Y$calY$ from the reduced divisor.
-The Stein construction yields $g_*OO_S=OO_{mathcal T}calT$,
+completion. This completion is mathcal YcalY and gives the map
+g:Stomathcal YcalY from the reduced divisor.
+The Stein construction yields g_*OO_S=OO_{mathcal T}calT,
and the line identities follow from those already established.
-Finally, start with the etale chart $S_Uto S$ obtained from $g$.
+Finally, start with the etale chart S_Uto S obtained from g.
Invariance of the etale site under nilpotent thickening extends
-this chart uniquely to $qS_Uto qS$, compatibly as $q$ varies.
-This construction does not use a map $qStomathcal Y$;calY$;
-lifting the map from $S$ is the task of the next section.
-The reduction $S_U$ is a scheme; its extension $qS_U$ is an
+this chart uniquely to qS_Uto qS, compatibly as q varies.
+This construction does not use a map qStomathcal Y;calY;
+lifting the map from S is the task of the next section.
+The reduction S_U is a scheme; its extension qS_U is an
algebraic space with trivial inertia. The chosen chart is
separated, the map to the root gerbe is finite, and that gerbe
-has finite diagonal, so $qS_U$ is separated over $X$.
-The finite representable map $Sto Dtimes_Pmathcal P$calP$
-and the etale map $S_Uto S$ show that the finite-type
-geometric fibers of $S_Uto X$ are zero-dimensional.
+has finite diagonal, so qS_U is separated over X.
+The finite representable map Sto Dtimes_Pmathcal PcalP
+and the etale map S_Uto S show that the finite-type
+geometric fibers of S_Uto X are zero-dimensional.
Nilpotent thickening changes neither geometric points nor
-fiber dimensions. The morphism $qS_Uto X$
+fiber dimensions. The morphism qS_Uto X
is therefore quasi-finite as well. Zariski's main theorem
-embeds $qS_U$ in a scheme finite over $X$, making it a
+embeds qS_U in a scheme finite over X, making it a
quasi-projective scheme as required.
@@ -1497,7 +1486,7 @@ Proposition~prop:signed-construction. We recall its data.
its tautological line bundle, and
[
Sxrightarrow{g}mathcal YcalYxrightarrow{p}mathcal P,calP,
- qquad h=pcirc g.
+ h=pcirc g.
]
Here (g) is representable and projective, and (p) is a smooth
projective bundle. The Stein factor of (h) is finite over
@@ -1510,7 +1499,7 @@ divisor (Ssubsetmathcal XcalX) has dimension (N=n-1). If
mathcal B:=calB:=
mathcal Hom_S(calHom_S(mathcal I_calI_{J/S},omega_S)
simeq OO_S(aS)=g^*C^a,
- qquad
+
OO_S(S)=g^*C,
for an integer (a>0). Off (T), the fixed ambient isomorphism
@@ -1597,9 +1586,9 @@ For (j:S_U∖ J_U↪ S_U), put
[
A^bullet_U=mathbf DbfD!(
j_!Q^H_{S_U∖ J_U}[N]),
- qquad
+
A_{0,U}={}^{p}!H^0A^bullet_U,
- qquad
+
M_U={}^{p}!H^1g_*A_{0,U}.
]
Here (mathbf DbfD) is Hodge-module duality, ({}^pH^i) denotes
@@ -1612,9 +1601,9 @@ These constructions commute with etale restriction. Thus the
The following identifications hold compatibly on the charts:
[
- F_{<0}A_{0,U}=0,qquad F_0A_{0,U}=mathcal B|_calB|_{S_U},
- qquad
- F_{<0}M_U=0,qquad
+ F_{<0}A_{0,U}=0, F_0A_{0,U}=mathcal B|_calB|_{S_U},
+
+ F_{<0}M_U=0,
F_0M_U=R^1g_*(mathcal B|_calB|_{S_U}).
]
In a smooth ambient embedding (S_U↪ V) of
@@ -1635,7 +1624,7 @@ graded de Rham complex with (mathcal I_calI_{J_U/S_U}).
Coherent duality and the maximal-Cohen--Macaulay property give
Gr^F_kDR(A^bullet_U)=0quad(k<0),
- qquad
+
Gr^F_0DR(A^bullet_U)=mathcal B|_calB|_{S_U}[0].
Here ([N]) cancels the dimension shift of the dualizing
@@ -1698,7 +1687,7 @@ projection. At filtration zero, only the top term of the relative
Spencer complex survives: it is the direct image of
(mathcal B|_calB|_{S_U}) on the graph. Projective strictness yields
[
- F_{<0}M_U=0,qquad
+ F_{<0}M_U=0,
F_0M_U=R^1g_*(mathcal B|_calB|_{S_U}),
]
and identifies this sheaf as a coherent subsheaf of the unfiltered
@@ -1878,7 +1867,7 @@ kills the conormal error.
Define
[
- E_{j,k}=g_*(I^j/I^{j+k}),qquad jgeq0,quad kgeq1.
+ E_{j,k}=g_*(I^j/I^{j+k}), jgeq0, kgeq1.
]
The first graded layer is
[
@@ -1955,7 +1944,7 @@ Write the overlap errors as
h_{j,lambda}-h_{i,lambda}
=e_{ij,lambda}y_i^k,
- qquad
+
y_j-y_i=eta_{ij}y_i^{k+1}.
Thus (D_k(t_lambda)) is represented by
@@ -1980,8 +1969,8 @@ structure sheaf. Realize the dualizing sheaves in support cohomology. Under
the trace inclusion
(omega_{S_U}↪omega_{(k+1)S_U}), we have
- b_j-b_i=-(a+k)eta_{ij}sigma,qquad
- y_i^kb_i=sigma,qquad y_i^{k+1}b_i=0.
+ b_j-b_i=-(a+k)eta_{ij}sigma,
+ y_i^kb_i=sigma, y_i^{k+1}b_i=0.
The first identity is the binomial expansion of
((1+eta_{ij}y_i^k)^{-(a+k)}): all quadratic terms vanish
@@ -2162,11 +2151,11 @@ transverse conormal frame.
# Compact null families and descent
-We now complete Theorem~thm:signed. Retain $0<v<n$,
-$r=v-1$, $d=n-1-r=n-v$, and the construction of
+We now complete Theorem~thm:signed. Retain 0<v<n,
+r=v-1, d=n-1-r=n-v, and the construction of
Proposition~prop:signed-construction. The formal lifts
first produce compact subvarieties outside the entire boundary,
-of dimension $n-v$, on which $L$ is numerically trivial.
+of dimension n-v, on which L is numerically trivial.
Their existence bounds the dimension of the nef reduction;
descent along that reduction then proves abundance.
@@ -2183,58 +2172,58 @@ neighborhood for all orders.
Assume the transition surjectivity of
Proposition~prop:formal-lifting. There is a dominating
-algebraic family of integral projective subvarieties of $X$
+algebraic family of integral projective subvarieties of X
of dimension
[
d=n-1-r=n-v
]
-whose general members avoid $D$ and on which $L$ is
+whose general members avoid D and on which L is
numerically trivial.
Proof.
We construct a formal deformation of a general boundary fiber,
-algebraize it over $C[[s]]$, and show that the resulting
-subvarieties belong to a family dominating $X$.
+algebraize it over C[[s]], and show that the resulting
+subvarieties belong to a family dominating X.
smallskipnoindentDeforming a boundary fiber.
-Choose a separated affine etale chart $Utomathcal Y$calY$
+Choose a separated affine etale chart Utomathcal YcalY
near a general point of a top-dimensional component of its
-Stein image $mathcal T_U$.calT_U$. After shrinking, $C$ is trivial,
-that component is smooth of pure dimension $r$, and $S_U$
-is flat over it. Since the image of $J$ in $P$ has dimension
-less than $r$, we may also arrange that $S_U$ is disjoint
-from $J_U$. Choose a closed point $t$ in this open set.
+Stein image mathcal T_U.calT_U. After shrinking, C is trivial,
+that component is smooth of pure dimension r, and S_U
+is flat over it. Since the image of J in P has dimension
+less than r, we may also arrange that S_U is disjoint
+from J_U. Choose a closed point t in this open set.
After a further shrinking, regular parameters
-$t_1,...,t_r$ on $mathcal T_U$calT_U$ cut out only $t$.
+t_1,...,t_r on mathcal T_UcalT_U cut out only t.
The fiber
[
F=S_{U,t}
]
-is projective and has pure dimension $d$. Along $F$, the
+is projective and has pure dimension d. Along F, the
pullbacks of the parameters form a regular sequence. The ideal
-sheaf $mathcal I_calI_{J/S}$ is maximal Cohen--Macaulay by
-Proposition~prop:signed-construction(ii). Since $S_U$ is
-disjoint from $J_U$, the etale pullback of this ideal sheaf is
-$mathcal I_calI_{J_U/S_U}=OO_{S_U}$; hence $S_U$ is
-Cohen--Macaulay. The quotient $F$ by the regular sequence is
+sheaf mathcal I_calI_{J/S} is maximal Cohen--Macaulay by
+Proposition~prop:signed-construction(ii). Since S_U is
+disjoint from J_U, the etale pullback of this ideal sheaf is
+mathcal I_calI_{J_U/S_U}=OO_{S_U}; hence S_U is
+Cohen--Macaulay. The quotient F by the regular sequence is
Cohen--Macaulay as well. We will use this property to control
the dimensions of the components after algebraization.
-Write $I=OO_{mathcal X}calX(-S)$. By
+Write I=OO_{mathcal X}calX(-S). By
Proposition~prop:formal-lifting, the transitions between
-the sheaves $g_*(I^j/I^{j+k})$ are surjective as the length
-$k$ increases. Their kernels are the coherent sheaves
-$g_*OO_{S_U} ⊗ C^{-j-k}$. Affineness of $U$ also gives
+the sheaves g_*(I^j/I^{j+k}) are surjective as the length
+k increases. Their kernels are the coherent sheaves
+g_*OO_{S_U} ⊗ C^{-j-k}. Affineness of U also gives
surjectivity on global sections. We can therefore lift the
parameters compatibly through all thickenings, and lift the
-chosen conormal frame to a compatible element $widetilde y$
-that formally generates $I$.
+chosen conormal frame to a compatible element widetilde y
+that formally generates I.
-Let $Z_q$ be the closed subscheme of $qS_U$ cut out by the
-$r$ lifted parameters. Regard it as a scheme over
+Let Z_q be the closed subscheme of qS_U cut out by the
+r lifted parameters. Regard it as a scheme over
[
- R_q=C[s]/(s^q),qquad s⟼widetilde y.
+ R_q=C[s]/(s^q), s⟼widetilde y.
]
The construction has the diagram
[
@@ -2245,26 +2234,26 @@ SpecC arrow[r,hook] & Spec R_q arrow[r,equal] & Spec R_q.
]
The middle vertical map is defined by the lifted normal parameter.
-The lifted base parameters cut out $Z_q$; they are not provided by a
+The lifted base parameters cut out Z_q; they are not provided by a
preexisting morphism of the ambient thickening to the base.
-The schemes $Z_q$ are compatible under reduction, have
-closed fiber $F$, and are flat over $R_q$. For flatness,
+The schemes Z_q are compatible under reduction, have
+closed fiber F, and are flat over R_q. For flatness,
the powers of the Cartier generator first identify the
-successive $s$-layers of $qS_U$ with $OO_{S_U}$. This
+successive s-layers of qS_U with OO_{S_U}. This
proves flatness before imposing the parameters. The local
flatness criterion then preserves flatness when we quotient
by lifts of the regular sequence on the closed fiber.
smallskipnoindentAlgebraization and boundary avoidance.
-The map $Z_qto XtimesSpec R_q$ is quasi-finite. It is
+The map Z_qto XtimesSpec R_q is quasi-finite. It is
also proper: its reduction is the map from the projective
-scheme $F$, and nilpotent thickenings do not change
+scheme F, and nilpotent thickenings do not change
properness of a finite-type morphism. Thus it is finite.
-Put $R=C[[s]]$. Projective Grothendieck existence
+Put R=C[[s]]. Projective Grothendieck existence
algebraizes the compatible finite algebra sheaves on
-$XtimesSpec R_q$ to a finite algebra on
-$X_R=XtimesSpec R$. Full faithfulness algebraizes
+XtimesSpec R_q to a finite algebra on
+X_R=XtimesSpec R. Full faithfulness algebraizes
their multiplication and unit
cite[Lemmas~30.24.1 and~30.24.3]{StacksExistence}.
The relative spectrum gives a finite morphism
@@ -2272,102 +2261,102 @@ The relative spectrum gives a finite morphism
Z→ X_R
]
with exactly the prescribed reductions. In particular,
-$Z$ is projective over $R$. Formal flatness gives
+Z is projective over R. Formal flatness gives
flatness along the closed fiber; on the generic fiber
-over $C((s))$ it is automatic. Hence $Z$ is flat over $R$.
+over C((s)) it is automatic. Hence Z is flat over R.
The images of the pole and coefficient-zero boundary parts
-miss $F$. Their inverse images in $Z$ are closed and
-proper over $R$, and must therefore be empty: any nonempty
-closed subset of a proper $R$-scheme specializes to the
-closed fiber. Recall also the exceptional divisor $E$ of the
-normalized blowup separating $S_+$ and $S_-$ in
-Proposition~prop:signed-construction. Its image in $X$
-lies in $Supp G_+capSupp G_-$, so it is already excluded
+miss F. Their inverse images in Z are closed and
+proper over R, and must therefore be empty: any nonempty
+closed subset of a proper R-scheme specializes to the
+closed fiber. Recall also the exceptional divisor E of the
+normalized blowup separating S_+ and S_- in
+Proposition~prop:signed-construction. Its image in X
+lies in Supp G_+capSupp G_-, so it is already excluded
by avoidance of the pole part.
-On the formal neighborhood from which the $Z_q$ were cut out,
+On the formal neighborhood from which the Z_q were cut out,
the root construction therefore identifies the divisor of the
-pulled-back section $s_0$ with $ell S$. Comparing its
+pulled-back section s_0 with ell S. Comparing its
root-line frame with the formal Cartier generator
-$widetilde y$ gives a frame in which this section is
-$widetilde y^ell$. Restricting to the $Z_q$ gives
+widetilde y gives a frame in which this section is
+widetilde y^ell. Restricting to the Z_q gives
compatible trivializations
[
N_0|_{widehatwidehat Z}ZsimeqOO_{widehatwidehat Z}Z,
- qquad s_0|_{widehatwidehat Z}Z=s^ell.
+ s_0|_{widehatwidehat Z}Z=s^ell.
]
Full faithfulness algebraizes this line-bundle trivialization
-and its section identity on $Z$; no algebraic map from $Z$
-to the root stack is needed. The generic fiber of $Z$ therefore misses
-the positive part of $D$ as well, and hence avoids all of
-$D$.
+and its section identity on Z; no algebraic map from Z
+to the root stack is needed. The generic fiber of Z therefore misses
+the positive part of D as well, and hence avoids all of
+D.
We next check the dimension assertion needed for the family.
-At a closed point $z$ of $F$, flatness makes $s$ a
-nonzerodivisor in $OO_{Z,z}$, and its quotient is the
-Cohen--Macaulay ring $OO_{F,z}$ of dimension $d$.
-Thus $OO_{Z,z}$ is Cohen--Macaulay of dimension $d+1$.
+At a closed point z of F, flatness makes s a
+nonzerodivisor in OO_{Z,z}, and its quotient is the
+Cohen--Macaulay ring OO_{F,z} of dimension d.
+Thus OO_{Z,z} is Cohen--Macaulay of dimension d+1.
These local rings are also equidimensional: a Cohen--Macaulay local ring
essentially of finite type over the excellent discrete
-valuation ring $R$ has this property.
-Every component of $Z$ meets $F$ by properness, and no
-component is contained in $F$ by flatness. The dimension
-formula now gives dimension $d$ for every component of the
-generic fiber. Its finite image in $X_{C((s))}$ therefore
-has pure dimension $d$. After a suitable finite extension of
-$C((s))$, choose a geometrically integral component of the
+valuation ring R has this property.
+Every component of Z meets F by properness, and no
+component is contained in F by flatness. The dimension
+formula now gives dimension d for every component of the
+generic fiber. Its finite image in X_{C((s))} therefore
+has pure dimension d. After a suitable finite extension of
+C((s)), choose a geometrically integral component of the
reduced base change of this image.
-smallskipnoindentA family dominating $X$.
-We have constructed compact subvarieties outside $D$.
+smallskipnoindentA family dominating X.
+We have constructed compact subvarieties outside D.
To obtain a dominating family, choose a positive component
-$D_i$ with image dimension $r$, together with a component
-of $S$ covering it. As the general chart and $t$ vary,
-the images of points of $F$ contain a dense open subset
-of $D_i$. We will show that each point in this open subset
+D_i with image dimension r, together with a component
+of S covering it. As the general chart and t vary,
+the images of points of F contain a dense open subset
+of D_i. We will show that each point in this open subset
lies in the closed evaluation image of an algebraic family
-of $d$-dimensional subvarieties disjoint from $D$.
+of d-dimensional subvarieties disjoint from D.
-Now consider all Hilbert schemes of $d$-dimensional
-subvarieties of $X$. The geometrically integral loci
-parametrizing members disjoint from $D$ have a countable
+Now consider all Hilbert schemes of d-dimensional
+subvarieties of X. The geometrically integral loci
+parametrizing members disjoint from D have a countable
stratification by integral parameter spaces with
irreducible universal families. Denote the irreducible
-closures of their evaluation images in $X$ by $W_alpha$.
-Fix a point $x$ in the dense open subset of $D_i$, and a
-point of a boundary fiber $F$ mapping to it. Since $Z$ has
+closures of their evaluation images in X by W_alpha.
+Fix a point x in the dense open subset of D_i, and a
+point of a boundary fiber F mapping to it. Since Z has
no vertical component, this point lies in the closure of a
-generic component. That component has dimension $d$ by
+generic component. That component has dimension d by
the preceding argument. Pass to a finite extension of the
discrete valuation field over which the reduced generic
-components are geometrically integral, and let $R'$ be the
+components are geometrically integral, and let R' be the
extended discrete valuation ring. Choose a generic component
after this base change whose closure contains a point over
-the chosen point of $F$. Flatness still excludes vertical
+the chosen point of F. Flatness still excludes vertical
components, so such a choice is possible. Its reduced
finite image is geometrically integral. Take the schematic
-closure of that image in $X_{R'}$.
+closure of that image in X_{R'}.
The structure sheaf of the closure has no uniformizer
torsion, so the closure is flat over the discrete valuation
ring and defines a morphism to the Hilbert scheme.
Its generic member lies in one of the strata above.
-The corresponding $W_alpha$ is closed and contains the
+The corresponding W_alpha is closed and contains the
generic image, hence also its special-fiber support and
-the point $x$. Thus every point of the chosen open subset
-of $D_i$ belongs to some $W_alpha$. The generic component
-and the index $alpha$ may depend on $x$.
+the point x. Thus every point of the chosen open subset
+of D_i belongs to some W_alpha. The generic component
+and the index alpha may depend on x.
-An irreducible variety over the uncountable field $C$
+An irreducible variety over the uncountable field C
cannot have a dense open covered by countably many proper
-closed subsets. Some $W_alpha$ therefore contains $D_i$.
-By definition it also contains points outside $D$.
-Since $D_i$ is a prime divisor in the integral variety
-$X$, the only proper irreducible closed subset containing
-it is $D_i$ itself. Hence $W_alpha=X$, and the family
-dominates $X$. On each member, avoidance of $D$ makes
-the rational section $s_0$ regular and nowhere zero.
-It trivializes $N_0$, proving that $L$ is numerically
+closed subsets. Some W_alpha therefore contains D_i.
+By definition it also contains points outside D.
+Since D_i is a prime divisor in the integral variety
+X, the only proper irreducible closed subset containing
+it is D_i itself. Hence W_alpha=X, and the family
+dominates X. On each member, avoidance of D makes
+the rational section s_0 regular and nowhere zero.
+It trivializes N_0, proving that L is numerically
trivial on that member.
@@ -2379,50 +2368,50 @@ that base, we first prove a statement for vertical divisors
using relative nefness and a fiber-dimension bound.
-Let $f:Yto B$ be a surjective projective morphism from a
+Let f:Yto B be a surjective projective morphism from a
normal integral variety to a smooth projective variety,
with geometrically connected integral generic fiber.
Suppose all fibers have dimension at most
-$dim Y-dim B$. Let $G_Y$ be a vertical $Q$-Cartier
+dim Y-dim B. Let G_Y be a vertical Q-Cartier
divisor that is relatively nef. Then
[
G_Y=f^*G_B
]
-for a $Q$-divisor $G_B$ on $B$.
+for a Q-divisor G_B on B.
Proof.
-When $B$ is a point, every vertical divisor is zero.
+When B is a point, every vertical divisor is zero.
When the relative dimension is zero, the fiber bound
-makes $f$ finite. Its geometrically integral connected
-generic fiber makes it birational, and normality of $B$
+makes f finite. Its geometrically integral connected
+generic fiber makes it birational, and normality of B
then makes it an isomorphism. The assertion follows in
both cases. We may therefore assume that the base
dimension and the relative dimension are positive.
The fiber bound forces every vertical prime divisor to
-map onto a prime divisor of $B$. For each such prime
-$P$, write its full pullback as $f^*P=sum_jm_jE_j$.
-Let $b_j$ be the coefficient of $G_Y$ along $E_j$, with
+map onto a prime divisor of B. For each such prime
+P, write its full pullback as f^*P=sum_jm_jE_j.
+Let b_j be the coefficient of G_Y along E_j, with
zero coefficients included, and assign coefficient
-$min_j(b_j/m_j)$ to $P$ in $G_B$. Only finitely many
+min_j(b_j/m_j) to P in G_B. Only finitely many
of these coefficients are nonzero. The residual divisor
[
R'=G_Y-f^*G_B
]
-is effective, vertical, $Q$-Cartier, and relatively nef;
+is effective, vertical, Q-Cartier, and relatively nef;
over each base prime, at least one component is absent
-from its support. It remains to prove that $R'=0$.
+from its support. It remains to prove that R'=0.
-Suppose $R'ne0$, and choose a base prime under its
-support. Intersect $B$ with general hyperplanes to
+Suppose R'ne0, and choose a base prime under its
+support. Intersect B with general hyperplanes to
obtain a complete-intersection curve meeting that prime
-at a general point; if $dim B=1$, use $B$ itself.
-Normal Bertini makes its inverse image in $Y$ normal.
+at a general point; if dim B=1, use B itself.
+Normal Bertini makes its inverse image in Y normal.
This inverse image is integral: generic geometric
integrality supplies the dominating component, and
the fiber bound rules out further vertical components.
-Intersect upstairs with $dim Y-dim B-1$ general
+Intersect upstairs with dim Y-dim B-1 general
very ample hyperplanes. The result is a normal integral
surface over the base curve, with geometrically connected
generic fiber. Over the chosen general base point,
@@ -2447,23 +2436,23 @@ its support.
Proof. [Completion of the proof of Theorem~thm:signed]
-The cases $v=0$ and $v=n$ are covered by
-Lemma~sg:numerical-boundary. For $0<v<n$,
+The cases v=0 and v=n are covered by
+Lemma~sg:numerical-boundary. For 0<v<n,
Propositions~prop:signed-construction,
prop:formal-lifting, and~prop:null-family
-give a dominating family of compact $L$-trivial
-subvarieties of dimension $d=n-v$.
+give a dominating family of compact L-trivial
+subvarieties of dimension d=n-v.
Apply the nef-reduction theorem to a Cartier multiple of
-$L$ cite[Theorem~2.1]{BauerEtAl2002}. The resulting almost
+L cite[Theorem~2.1]{BauerEtAl2002}. The resulting almost
holomorphic rational map has connected fibers. The
-divisor $L$ is numerically trivial on its compact general
+divisor L is numerically trivial on its compact general
fibers and has positive degree on every noncontracted
-curve through a very general point of $X$. Write $b$
-for the base dimension. A compact $L$-trivial $d$-fold
+curve through a very general point of X. Write b
+for the base dimension. A compact L-trivial d-fold
through such a point must be contracted: otherwise,
general ample slices through the point would produce a
-noncontracted curve of $L$-degree zero. Consequently
+noncontracted curve of L-degree zero. Consequently
[
ble n-d=v.
]
@@ -2478,18 +2467,18 @@ This gives projective morphisms
B &
]
-with $pi$ birational, $B$ smooth, $Y$ normal, and
-geometrically connected integral generic fiber of $f$.
-Every fiber has dimension at most $n-b$. Indeed,
+with pi birational, B smooth, Y normal, and
+geometrically connected integral generic fiber of f.
+Every fiber has dimension at most n-b. Indeed,
flatness and generic integrality keep the flat main
transform integral after the base modification, and
finite normalization preserves the bound on fiber
dimension. This bound is the needed hypothesis;
normalization need not preserve flatness.
-We claim that $pi^*D$ has no horizontal component.
-Restrict to a very general fiber of $f$. There,
-$pi^*L$ is numerically trivial and $pi^*(L-cD)$
+We claim that pi^*D has no horizontal component.
+Restrict to a very general fiber of f. There,
+pi^*L is numerically trivial and pi^*(L-cD)
restricts to a pseudo-effective class. To justify the
second assertion, choose a sequence of effective approximants with
ample errors tending to zero. Shrink to a flat open of the base with
@@ -2500,32 +2489,32 @@ proper closed, by upper semicontinuity of fiber dimension.
Avoid the countably many such bad closed loci in the base.
Every approximant then restricts effectively to a very general fiber,
and passage to the limit gives pseudo-effectivity of the restriction.
-The resulting class is $-cpi^*D$.
+The resulting class is -cpi^*D.
A negative nonzero effective divisor cannot be
pseudo-effective on a projective variety, as its
intersection with an ample power shows. The restriction
-of $pi^*D$ must therefore be zero. This also excludes
-$b=0$, which would force $D=0$ and contradict $v>0$.
+of pi^*D must therefore be zero. This also excludes
+b=0, which would force D=0 and contradict v>0.
-It follows that $G_Y=pi^*G$ is vertical. It is also
-relatively nef because $G_Ysim_{Q}pi^*L$.
+It follows that G_Y=pi^*G is vertical. It is also
+relatively nef because G_Ysim_{Q}pi^*L.
Lemma~sg:vertical-descent now gives
[
pi^*Lsim_{Q}f^*G_B
]
-for a $Q$-divisor $G_B$ on the smooth base. Every
-curve in $B$ is dominated by a curve in $Y$, where
-the pullback has nonnegative degree; hence $G_B$ is
+for a Q-divisor G_B on the smooth base. Every
+curve in B is dominated by a curve in Y, where
+the pullback has nonnegative degree; hence G_B is
nef. Numerical dimension is unchanged by these pullbacks;
the intersection formula gives
[
v=nu(X,L)=nu(Y,pi^*L)=nu(B,G_B)le b.
]
-Together with $ble v$, this yields $b=v=nu(B,G_B)$.
-Since $G_B$ is nef on the $b$-dimensional base, it follows
-that $G_B^b>0$, so $G_B$ is big. Pullback of
-sections then gives $kappa(X,L)ge b=v$. The
-inequality $kappa(X,L)lenu(X,L)$ for nef divisors
+Together with ble v, this yields b=v=nu(B,G_B).
+Since G_B is nef on the b-dimensional base, it follows
+that G_B^b>0, so G_B is big. Pullback of
+sections then gives kappa(X,L)ge b=v. The
+inequality kappa(X,L)lenu(X,L) for nef divisors
provides the reverse bound and proves abundance.
@@ -2538,7 +2527,7 @@ Assumption~mmp:lower-models, and suppose that a smooth
projective (n)-fold (X), with (n>0), satisfies
K_X is pseudo-effective,
- qquad kappa(X,K_X)=-∞.
+ kappa(X,K_X)=-∞.
We first restrict its moving proper subvarieties and positive canonical
currents, and then exclude covering curves of bounded genus and normalized
@@ -2685,7 +2674,7 @@ fiber directions zero as well. A generically etale evaluation
transfers this assertion to the target. Annihilation passes to
brackets by the identities
[
- mathcal L_vcalL_v T=d(iota_vT)+iota_v(dT),qquad
+ mathcal L_vcalL_v T=d(iota_vT)+iota_v(dT),
iota_{[v,w]}T=mathcal L_v(calL_v(iota_wT)-iota_w(mathcal L_vT).calL_vT).
]
Pullbacks by dominant morphisms and restrictions to almost every
@@ -2966,7 +2955,7 @@ otherwise a prime in such a pullback that dominates a divisor of (Y) would
have center of codimension at least two on (B), contrary
to the property just established.
-smallskip
+
noindentDescent of the current.
Let (T) also denote the pulled-back current on (W).
Shrink to a smooth open (B^0) so that (h) is smooth
@@ -3029,7 +3018,7 @@ disappear and the remaining terms are effective, so
The horizontal components removed during descent contribute
effective divisors to the same difference.
-smallskip
+
noindentA negative canonical direction on the base.
Choose a nonzero supporting functional (alpha) of the
pseudo-effective cone at (K_Y), and put
@@ -3099,7 +3088,7 @@ so (kappa(Y,P)geq1). Corollary~ex:supporting now gives
(0<alpha(P)=-K_Bcdoteta), proving
(ex:negative-base).
-smallskip
+
noindentRational curves and an exact zero gap.
Apply the multiplier approximation to (S) on (B), and
principalize (mathcal J(kS)calJ(kS)) by (r_k:B_kto B).
@@ -3114,7 +3103,7 @@ Since (etacdot{S}=0) and (L_k-k{S}) stays in a
bounded set, this gives the first estimate below. The second
uses the fact that (eta_k) kills exceptional divisors:
[
- 0leq P_kcdoteta_k=O(k^{-1}),qquad
+ 0leq P_kcdoteta_k=O(k^{-1}),
K_{B_k}cdoteta_k=K_Bcdoteta<0.
]
Fix an ample divisor (H) on (B), and write
@@ -3131,7 +3120,7 @@ Approximate (eta_k) by a positive sum
of covering-curve classes,
closely enough that
[
- Q_kcdotgamma_kleq C/k,qquad
+ Q_kcdotgamma_kleq C/k,
-K_{B_k}cdotgamma_kgeqdelta/2
]
for a constant (C) independent of (k). Keep only terms
@@ -3152,7 +3141,7 @@ uncountability give a covering family of such rational curves,
denoted (R_k). Since (Q_k-P_k-k^{-1}r_k^*H) is ample
and both (P_k) and (r_k^*H) are nef, we obtain
- P_kcdot R_k=O(k^{-1}),qquad
+ P_kcdot R_k=O(k^{-1}),
Hcdot r_{k*}R_k=O(1).
Shrink each parameter space so that the curve class and its
@@ -3263,7 +3252,7 @@ Fix a positive current (T) on a smooth resolution in the
pullback class of (K_Y). On a common smooth resolution of
(Ydashrightarrow Y_t), write
[
- p^*K_Y=p_t^*K_t+J_t,qquad J_tgeq0,
+ p^*K_Y=p_t^*K_t+J_t, J_tgeq0,
]
where (J_t) is exceptional over (Y_t).
Then (nu_E(T)geqcoeff_E J_t) for every
@@ -3317,7 +3306,7 @@ There are no sequences (t_j↓0) and covering
families of curves on (Y_{t_j}), with smooth projective
domains (C_j), for which both
[
- g(C_j),qquad
+ g(C_j),
frac{(K_{t_j}+t_jA_{t_j})cdot C_j}{mu_{t_j}}
]
are bounded by a fixed constant. Degrees mean degrees on
@@ -3366,7 +3355,7 @@ discrepancies in (m^{-1}Z). In particular
remaining ramification coefficients are nonnegative.
On the general parameter fiber,
[
- K_{U_t}cdot C_t=2g(C_t)-2,qquad
+ K_{U_t}cdot C_t=2g(C_t)-2,
q_t^*K_Ycdot C_tgeq0.
]
The latter inequality follows from pseudoeffectivity and
@@ -3492,7 +3481,7 @@ Take a small projective (Q)-factorialization
(p_i:S'_ito S_i), and put
[
D_i=K_{S'_i}+B_i=p_i^*((K_{V_i}+Delta_i)|_{S_i}),
- qquad C_i=p_i^*(A_i|_{S_i}).
+ C_i=p_i^*(A_i|_{S_i}).
]
Adjunction gives a dlt pair ((S'_i,B_i)), with (B_i,C_igeq0),
and ((S'_i,B_i+lambda_i C_i)) is lc. The scaling divisor
@@ -3510,8 +3499,8 @@ connected fibers give descent. Restricting and pulling back yields
[
D_i+lambda_i C_isim_{Q}0/T_i.
]
-The descended divisor on $T_i$ is nef: its pullback to $S'_i$
-is the pullback of the restriction of the globally nef $L_i$.
+The descended divisor on T_i is nef: its pullback to S'_i
+is the pullback of the restriction of the globally nef L_i.
By cite[Theorem~1.1(3)]{Birkar2012Special}, a (D_i)-LMMP
over (T_i) with scaling of a fresh ample divisor terminates.
@@ -3528,16 +3517,16 @@ The next point is that these finite relative programs concatenate
into an absolute program with scaling of the transforms of (C_1).
Throughout the (i)-th piece, (D_i+lambda_i C_i) remains
the pullback of a nef divisor on (T_i), hence is globally nef.
-Here $D_i$ and $C_i$ also denote their transforms along this
-relative program. A contracted negative ray $R$ satisfies
+Here D_i and C_i also denote their transforms along this
+relative program. A contracted negative ray R satisfies
[
(D_i+lambda_i C_i)cdot R=0,
- qquad D_icdot R<0.
+ D_icdot R<0.
]
-It follows that $C_icdot R>0$. Every smaller coefficient of
-$C_i$ leaves this ray negative, while $D_i+lambda_i C_i$ is
+It follows that C_icdot R>0. Every smaller coefficient of
+C_i leaves this ray negative, while D_i+lambda_i C_i is
globally nef. Thus the absolute scaling threshold at that step
-is exactly $lambda_i$.
+is exactly lambda_i.
The relative cone is a face of the absolute cone, cut out by
the pullback of an ample divisor on the projective base (T_i);
its extremal rays are therefore absolute extremal rays.
@@ -3546,23 +3535,23 @@ Rescaling the initial scaling divisor by (lambda_1), if
necessary, makes its sum with the initial boundary lc and nef.
If the concatenation were infinite, its positive scaling numbers
-would tend to zero. Fix a late scaling value $lambda_j$.
-On every earlier ray $R$ contracted at threshold $lambda_i$,
+would tend to zero. Fix a late scaling value lambda_j.
+On every earlier ray R contracted at threshold lambda_i,
the calculation above gives
[
(D_i+lambda_j C_i)cdot R
=(lambda_j-lambda_i)C_icdot Rle0.
]
Thus all preceding steps are nonpositive for the adjoint with
-this fixed coefficient $lambda_j$. On a common resolution of
-the initial and the $j$-th models, the pullback of
-$D_1+lambda_j C_1$ equals the pullback of the late nef divisor
-$D_j+lambda_j C_j$ plus an effective comparison divisor.
+this fixed coefficient lambda_j. On a common resolution of
+the initial and the j-th models, the pullback of
+D_1+lambda_j C_1 equals the pullback of the late nef divisor
+D_j+lambda_j C_j plus an effective comparison divisor.
Push this equality down to the initial model. The pushforward
of the nef pullback is pseudo-effective, as is the effective
-comparison term, so $D_1+lambda_j C_1$ is pseudo-effective.
-Closedness of the pseudo-effective cone and $lambda_jto0$
-show that $D_1$ is pseudo-effective.
+comparison term, so D_1+lambda_j C_1 is pseudo-effective.
+Closedness of the pseudo-effective cone and lambda_jto0
+show that D_1 is pseudo-effective.
Assumption~mmp:lower-models therefore gives an absolute
log minimal model for that initial pair. The concatenation
terminates by cite[Theorem~1.9(iii)]{Birkar2012Special}, since
@@ -3713,7 +3702,7 @@ counterexample. Resolve the generically finite rational map from
that abelian variety to (X):
on a smooth model (U), ramification gives
[
- K_U=f^*K_X+R,qquad Rge0.
+ K_U=f^*K_X+R, Rge0.
]
The canonical class of (U) also has the effective representative
(E_A) exceptional over the abelian variety: choosing a nowhere-zero
@@ -3751,17 +3740,17 @@ minimal models, nor logarithmic subadditivity.
Section~sec:jets applies them to moving components produced by
failure of the scalar or two-slot jet estimates.
-All varieties in the moving-center arguments are over $C$.
+All varieties in the moving-center arguments are over C.
A rational Cartier divisor is a rational divisor some positive
-integral multiple of which is Cartier. A requirement that $kP$
-be Cartier means that $kP$ is an actual integral Cartier divisor.
+integral multiple of which is Cartier. A requirement that kP
+be Cartier means that kP is an actual integral Cartier divisor.
For a nef
class on an integral subvariety, intersections are computed after
pullback to its normalization and a resolution. For a linear
subseries of a line bundle, its base ideal is the image of its
evaluation map after tensoring by the inverse line bundle.
-``Isolated at the generic point of $V$'' means that this ideal
-is primary for the maximal ideal of $mathcal O_calO_{Z,eta_V}$.
+``Isolated at the generic point of V'' means that this ideal
+is primary for the maximal ideal of mathcal O_calO_{Z,eta_V}.
All powers of ideals below are ordinary powers.
## Numerical subadjunction on the center
@@ -3774,80 +3763,80 @@ auxiliary pair to be log canonical away from the generic center.
[Numerical generic subadjunction]
-Let $Z$ be a projective $Q$-factorial klt variety over $C$, let
-$Θgeq0$ be a rational divisor, and let $Vsubset Z$ be an
-integral positive-dimensional subvariety. Suppose that $(Z,Θ)$
-is lc at the generic point of $V$ and that a divisor of log
-discrepancy zero has center $V$. Let $f=dim V$, let $P$ be a nef
-rational Cartier divisor on $V$, and let $rho:V^*to V$ be a smooth
+Let Z be a projective Q-factorial klt variety over C, let
+Θgeq0 be a rational divisor, and let Vsubset Z be an
+integral positive-dimensional subvariety. Suppose that (Z,Θ)
+is lc at the generic point of V and that a divisor of log
+discrepancy zero has center V. Let f=dim V, let P be a nef
+rational Cartier divisor on V, and let rho:V^*to V be a smooth
projective resolution, factoring through the normalization. Then
K_{V^*}cdot(rho^*P)^{f-1}
leq (K_Z+Θ)|_Vcdot P^{f-1}.
-The right side is the intersection on $V$ of the restricted
-rational Cartier class with $P^{f-1}$. No $Q$-Gorenstein hypothesis
-on the normalization of $V$ is required.
+The right side is the intersection on V of the restricted
+rational Cartier class with P^{f-1}. No Q-Gorenstein hypothesis
+on the normalization of V is required.
Proof.
Apply the dlt blowup theorem for an effective rational divisor
cite[Theorem~2.10]{FujinoHashizume2023}. It gives a projective
-birational morphism $p:Qto Z$ with $Q$ $Q$-factorial and
+birational morphism p:Qto Z with Q Q-factorial and
p^*(K_Z+Θ)=K_Q+B_Q+E_Q,
-where $(Q,B_Q)$ is dlt and $E_Qgeq0$ is the surplus over the
+where (Q,B_Q) is dlt and E_Qgeq0 is the surplus over the
non-lc locus. In this construction the coefficients of the strict
boundary are truncated at one and the relevant extracted divisors
-are reduced; the coefficients left over form $E_Q$. The surplus
-is absent over the open where $(Z,Θ)$ is lc. In particular it
-is absent over a neighborhood of the generic point of $V$.
+are reduced; the coefficients left over form E_Q. The surplus
+is absent over the open where (Z,Θ) is lc. In particular it
+is absent over a neighborhood of the generic point of V.
-There are lc strata of $(Q,B_Q)$ mapping onto $V$. Indeed over that
+There are lc strata of (Q,B_Q) mapping onto V. Indeed over that
lc neighborhood the construction is crepant, and the center of a
log-discrepancy-zero divisor on the dlt model is an lc stratum.
-Choose a stratum $S$ minimal among those mapping onto $V$.
-Iterated dlt adjunction makes $S$ normal and gives an effective
-rational dlt different on $S$; its lc centers are precisely the
-smaller ambient strata in $S$
-[cite]. The divisor $E_Q$ does not contain $S$. Since
-$Q$ is $Q$-factorial, a Cartier multiple of $E_Q$ restricts to an
-effective Cartier divisor on $S$. Adding that restriction to the
-different gives an effective rational divisor $B_S$ such that
+Choose a stratum S minimal among those mapping onto V.
+Iterated dlt adjunction makes S normal and gives an effective
+rational dlt different on S; its lc centers are precisely the
+smaller ambient strata in S
+[cite]. The divisor E_Q does not contain S. Since
+Q is Q-factorial, a Cartier multiple of E_Q restricts to an
+effective Cartier divisor on S. Adding that restriction to the
+different gives an effective rational divisor B_S such that
K_S+B_Ssim_{Q}(p|_S)^*((K_Z+Θ)|_V).
-Over the generic point of $V$ this pair is klt: the surplus is
+Over the generic point of V this pair is klt: the surplus is
absent there, and a non-klt center of the different dominating
-$V$ would be a smaller stratum, contradicting the choice of $S$.
+V would be a smaller stratum, contradicting the choice of S.
-Factor the map through the normalization $V^nu$ and its Stein
+Factor the map through the normalization V^nu and its Stein
factorization:
[
Sxrightarrow{q}V'xrightarrow{eta}V^nuxrightarrow{nu}V.
]
-Here $q$ has connected fibers and $eta$ is finite. Put
-$D_V=nu^*((K_Z+Θ)|_V)$ and
-$D'=eta^*D_V$. We apply the canonical bundle formula on the
-connected-fiber base $V'$, then push the resulting intersection
-inequality through the finite map $eta$. Its ramification will
+Here q has connected fibers and eta is finite. Put
+D_V=nu^*((K_Z+Θ)|_V) and
+D'=eta^*D_V. We apply the canonical bundle formula on the
+connected-fiber base V', then push the resulting intersection
+inequality through the finite map eta. Its ramification will
contribute an effective term. The final comparison is between the
-canonical cycle on $V^nu$ and the canonical divisor on $V^*$.
+canonical cycle on V^nu and the canonical divisor on V^*.
We verify the klt-trivial-fibration hypotheses for
-$q:(S,B_S)to V'$. The generic pair is sub-klt, and
+q:(S,B_S)to V'. The generic pair is sub-klt, and
Equation~(common:subadjunction-stratum) is the required rational-linear
pullback expression. For the rank condition, take a log resolution
-over the generic point of $V'$. Since the boundary on $S$ is
+over the generic point of V'. Since the boundary on S is
effective and the generic pair is klt, the round-up of its
discrepancy divisor has coefficient zero along every strict
boundary component and has nonnegative coefficients only on
-exceptional divisors. Over this dense klt open, its pushforward to the normal $S$ is
-$cO_S$: a rational function allowed poles only over exceptional
+exceptional divisors. Over this dense klt open, its pushforward to the normal S is
+cO_S: a rational function allowed poles only over exceptional
centers extends across codimension at least two. Connectedness
in the Stein factorization therefore gives generic rank one for
the required discrepancy round-up pushforward. This verifies the
@@ -3860,63 +3849,63 @@ the form recalled by Fujino--Gongyo, now gives
D'sim_{Q}K_{V'}+B_{V'}+M_{V'},
-where $M$ is a b-nef moduli divisor
+where M is a b-nef moduli divisor
cite[Definition~3.1 and Theorem~3.5]{FujinoGongyoModuli2014}.
The definition requires sub-klt over the generic point; it does
not require the pair to be globally sub-lc. Thus the possible
-non-lc fibers caused by $E_Q|_S$ are permitted.
+non-lc fibers caused by E_Q|_S are permitted.
-The discriminant $B_{V'}$ on this initial base is effective. To
-check this at a prime divisor $Fsubset V'$, work over its generic
+The discriminant B_{V'} on this initial base is effective. To
+check this at a prime divisor Fsubset V', work over its generic
point, where the base is regular. A component of its inverse
-image on $S$ has multiplicity $m_Fgeq1$ and nonnegative boundary
-coefficient $b_F$. Its discrepancy bounds the generic lc threshold
-$t_F$ from above by
+image on S has multiplicity m_Fgeq1 and nonnegative boundary
+coefficient b_F. Its discrepancy bounds the generic lc threshold
+t_F from above by
[
t_Fleq(1-b_F)/(m_F)leq1.
]
-Thus the discriminant coefficient $1-t_F$ is nonnegative. If the
-fiber is already non-lc, $t_F$ can be negative; this only increases
+Thus the discriminant coefficient 1-t_F is nonnegative. If the
+fiber is already non-lc, t_F can be negative; this only increases
that coefficient.
-Let $P'=eta^*nu^*P$. On a higher projective base
-$mu:widehatwidehat V'V'to V'$ where the moduli divisor is nef, its
-trace satisfies $M_{V'}=mu_*M_{widehatwidehat V'}V'$. The projection
+Let P'=eta^*nu^*P. On a higher projective base
+mu:widehatwidehat V'V'to V' where the moduli divisor is nef, its
+trace satisfies M_{V'}=mu_*M_{widehatwidehat V'}V'. The projection
formula gives
[
M_{V'}cdot(P')^{f-1}
=M_{widehatwidehat V'}V'cdot(mu^*P')^{f-1}geq0.
]
Only this intersection is asserted to be nonnegative; the trace
-$M_{V'}$ need not itself be nef. Likewise
-$B_{V'}cdot(P')^{f-1}geq0$ by effectivity and nefness.
+M_{V'} need not itself be nef. Likewise
+B_{V'}cdot(P')^{f-1}geq0 by effectivity and nefness.
All intersections with Weil divisors here mean their cycle
intersections with rational Cartier powers, so they are defined
even when the separate canonical or discriminant divisor is not
-$Q$-Cartier.
+Q-Cartier.
-Choose compatible canonical Weil divisors. Since $eta$ is finite
+Choose compatible canonical Weil divisors. Since eta is finite
and separable, the codimension-one ramification formula, followed
by pushforward, is
[
eta_*K_{V'}=(degeta)K_{V^nu}+eta_*R_eta,
- qquad R_etageq0.
+ R_etageq0.
]
This identity of Weil cycles uses the discrete valuation rings at
-generic prime divisors and does not require $K_{V^nu}$ to be
-$Q$-Cartier. Intersect Equation~(common:subadjunction-cbf) with
-$(P')^{f-1}$ and discard the nonnegative discriminant and moduli
-intersections. Pushforward through $eta$ and the nonnegative
+generic prime divisors and does not require K_{V^nu} to be
+Q-Cartier. Intersect Equation~(common:subadjunction-cbf) with
+(P')^{f-1} and discard the nonnegative discriminant and moduli
+intersections. Pushforward through eta and the nonnegative
ramification intersection then give
[
(degeta)D_Vcdot(nu^*P)^{f-1}
geq(degeta)K_{V^nu}cdot(nu^*P)^{f-1}.
]
-Finally the pushforward of $K_{V^*}$ to $V^nu$ is its compatible
+Finally the pushforward of K_{V^*} to V^nu is its compatible
canonical cycle. All additional divisors on the resolution have
image of codimension at least two, so their intersection with
-$f-1$ pulled-back Cartier powers is zero. Dividing by
-$degeta$ proves Equation~(common:subadjunction-inequality). When $f=1$
+f-1 pulled-back Cartier powers is zero. Dividing by
+degeta proves Equation~(common:subadjunction-inequality). When f=1
there are no birational exceptional divisors on the normal curve,
and the same calculation applies directly.
@@ -3929,91 +3918,91 @@ degree of the center and the coefficient of that boundary.
[A base component of high multiplicity]
-Let $Z$ be a projective $Q$-factorial klt variety of dimension $d$,
-let $P$ be a nef rational Cartier divisor, and let $k>0$ be an
-integer for which $kP$ is Cartier. Let
-$0nemathcal HcalHsubset H^0(Z,cO_Z(kP))$ 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
-$eta_Vin Z_{rm sm}$. Write $a=d-f$ and
-$mathfrak m=frakm=mathfrak m_frakm_{cO_{Z,eta_V}}$. If
+Let Z be a projective Q-factorial klt variety of dimension d,
+let P be a nef rational Cartier divisor, and let k>0 be an
+integer for which kP is Cartier. Let
+0nemathcal HcalHsubset H^0(Z,cO_Z(kP)) 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_{rm sm}. Write a=d-f and
+mathfrak m=frakm=mathfrak m_frakm_{cO_{Z,eta_V}}. If
[
mathfrak bfrakbcO_{Z,eta_V}subsetmathfrak m^hfrakm^h
- qquad(hinZ_{>0}),
+ (hinZ_{>0}),
]
then
P^fcdot Vleq (k/h)^aP^d.
-There are a rational number $c$ with $0<cleq ak/h$ and an
-effective rational divisor $Θsim_{Q}cP$ such that
-$(Z,Θ)$ is lc at $eta_V$ and has an lc place with center
-$V$. If $f>0$, on a smooth resolution $V^*$ one consequently has
+There are a rational number c with 0<cleq ak/h and an
+effective rational divisor Θsim_{Q}cP such that
+(Z,Θ) is lc at eta_V and has an lc place with center
+V. If f>0, on a smooth resolution V^* one consequently has
K_{V^*}cdot P^{f-1}
leq (K_Z+cP)|_Vcdot P^{f-1}.
-The degree estimate includes $f=0$; the canonical estimate is
-asserted only for positive-dimensional $V$.
+The degree estimate includes f=0; the canonical estimate is
+asserted only for positive-dimensional V.
Proof.
-The local ring $R=cO_{Z,eta_V}$ is regular of dimension $a$.
-Isolation means that $mathfrak bR$frakbR$ is $mathfrak m$-primary.frakm$-primary.
-Choose $a$ general sections of $mathcal H$.calH$. Their local equations
-form a system of parameters in $R$, since an $mathfrak m$-primaryfrakm$-primary
-ideal is contained in no smaller prime. Let $mathfrak q$frakq$ be the
-parameter ideal they generate. It is contained in $mathfrak m^h$.frakm^h$.
+The local ring R=cO_{Z,eta_V} is regular of dimension a.
+Isolation means that mathfrak bRfrakbR is mathfrak m-primary.frakm-primary.
+Choose a general sections of mathcal H.calH. Their local equations
+form a system of parameters in R, since an mathfrak m-primaryfrakm-primary
+ideal is contained in no smaller prime. Let mathfrak qfrakq be the
+parameter ideal they generate. It is contained in mathfrak m^h.frakm^h.
Regularity and the multiplicity comparison give
[
length(R/mathfrak q)frakq)
=e(mathfrak q,R)frakq,R)geq e(mathfrak m^h,R)=h^a.frakm^h,R)=h^a.
]
-This is the generic intersection multiplicity along $V$.
+This is the generic intersection multiplicity along V.
The global intersection can be performed without assuming that
the series has no other base components. Before each cut, discard
the components of the current effective cycle that lie in the
base locus. Before the final cut their dimensions are greater
-than $f$, so none can contain $V$: otherwise $V$ would not be a
+than f, so none can contain V: otherwise V would not be a
base-locus component. Discarding them therefore leaves the
-calculation at $eta_V$ unchanged. A general next section meets
-the remaining components properly. Nefness of $P$ shows that
+calculation at eta_V unchanged. A general next section meets
+the remaining components properly. Nefness of P shows that
discarding effective components can only decrease their total
-$P$-degree. After $a$ cuts the surviving effective cycle has
-$P$-degree at most $k^aP^d$, and contains $V$ with multiplicity at
-least $h^a$. This proves Equation~(common:base-component-degree).
+P-degree. After a cuts the surviving effective cycle has
+P-degree at most k^aP^d, and contains V with multiplicity at
+least h^a. This proves Equation~(common:base-component-degree).
-Let $lambda=lct_{eta_V}(mathfrak b)$frakb)$ be the generic ideal
+Let lambda=lct_{eta_V}(mathfrak b)frakb) be the generic ideal
threshold. It is positive and rational. The valuation from
-blowing up the smooth generic center has log discrepancy $a$
-and ideal order at least $h$, so
+blowing up the smooth generic center has log discrepancy a
+and ideal order at least h, so
[
0<lambdaleq a/h.
]
-Because $mathfrak bR$frakbR$ is primary, a divisor computing this
-threshold has center $V$. We realize the ideal threshold by a
+Because mathfrak bRfrakbR is primary, a divisor computing this
+threshold has center V. We realize the ideal threshold by a
divisor rather than apply subadjunction to an ideal formally.
-Take a log resolution of $Z$ and $mathfrak b$,frakb$, so that the
-pulled-back system has fixed divisor $F$ and a basepoint-free
+Take a log resolution of Z and mathfrak b,frakb, so that the
+pulled-back system has fixed divisor F and a basepoint-free
moving part. Choose sufficiently many general members
-$D_1,...,D_N$ of $mathcal H$calH$ and put
+D_1,...,D_N of mathcal HcalH and put
[
Θ=(lambda)/(N)(D_1+...+D_N)
- sim_{Q}cP,qquad c=klambda.
+ sim_{Q}cP, c=klambda.
]
-On the resolution the fixed contribution is $lambda F$.
-The moving contributions have coefficients $lambda/N<1$ and,
+On the resolution the fixed contribution is lambda F.
+The moving contributions have coefficients lambda/N<1 and,
by general choice, meet the fixed resolution divisor with simple
-normal crossings on the open in question. Thus $(Z,Θ)$ is
-lc at the generic point of $V$ and has the same lc place there
+normal crossings on the open in question. Thus (Z,Θ) is
+lc at the generic point of V and has the same lc place there
as the ideal threshold. Global log canonicity is unnecessary.
-For $f>0$, Lemma~common:numerical-subadjunction gives
+For f>0, Lemma~common:numerical-subadjunction gives
Equation~(common:base-component-canonical), and
-$c=klambdaleq ak/h$.
+c=klambdaleq ak/h.
## Differentiating the full moving jet kernel
@@ -4030,49 +4019,49 @@ and primary-ideal details needed here are included.
[Moving multiplicity]
-Let $Z$ be an integral projective variety of dimension $d$, and
-let $P$ be a nef, big, semiample rational Cartier divisor. Fix an integer $k$
-with $kP$ Cartier and positive real numbers
+Let Z be an integral projective variety of dimension d, and
+let P be a nef, big, semiample rational Cartier divisor. Fix an integer k
+with kP Cartier and positive real numbers
[
tau_0<tau_1<...<tau_d,
qquadtau_{i+1}-tau_i=delta>0.
]
-For a smooth point $x$, let $mathcal H_i(x)$calH_i(x)$ be the entire
-subspace of sections of $kP$ vanishing at $x$ to order at least
-$lceil ktau_irceil$. Suppose that for very general $x$ all these
-subspaces are nonzero and the base locus of $mathcal H_0(x)$calH_0(x)$
-has a positive-dimensional component through $x$.
-If $kdeltageq4$, then, after an algebraic parameter extension
+For a smooth point x, let mathcal H_i(x)calH_i(x) be the entire
+subspace of sections of kP vanishing at x to order at least
+lceil ktau_irceil. Suppose that for very general x all these
+subspaces are nonzero and the base locus of mathcal H_0(x)calH_0(x)
+has a positive-dimensional component through x.
+If kdeltageq4, then, after an algebraic parameter extension
and restriction to nonempty opens, there are an irreducible
-parameter space $T$, a fixed adjacent pair of levels, and a
-family of integral subvarieties $V_t$ common to the two base
+parameter space T, a fixed adjacent pair of levels, and a
+family of integral subvarieties V_t common to the two base
loci such that the incidence
[
Gamma={(t,z):zin V_t}subset Ttimes Z
]
-dominates $Z$. Each general $V_t$ has dimension in $[1,d-1]$,
+dominates Z. Each general V_t has dimension in [1,d-1],
is an isolated component of the higher base locus at its generic
point, and its entire higher-series base ideal is contained
there in
- mathcal I_calI_{V_t}^{h},qquad h=lceil kdelta/2rceil,
- qquad k/hleq2/delta.
+ mathcal I_calI_{V_t}^{h}, h=lceil kdelta/2rceil,
+ k/hleq2/delta.
These are ordinary local ideal powers in the smooth ambient open.
-The incidence base ideal lies in $mathcal I_calI_Gamma^h$ on a dense
-open of $Gamma$. If, in addition, $Z$ is $Q$-factorial and klt,
-then for a general member, with $f=dim V_t$, one has
+The incidence base ideal lies in mathcal I_calI_Gamma^h on a dense
+open of Gamma. If, in addition, Z is Q-factorial and klt,
+then for a general member, with f=dim V_t, one has
P^fcdot V_t&le(2/delta)^{d-f}P^d,
K_{V_t^*}P^{f-1}&le(K_Z+cP)|_{V_t}P^{f-1},
- qquad 0<cle2(d-f)/delta .
+ 0<cle2(d-f)/delta .
-In particular the weaker bound $cle2d/delta$ also holds.
-Positivity of $P^fcdot V_t$ follows if $V_t$ meets the open
+In particular the weaker bound cle2d/delta also holds.
+Positivity of P^fcdot V_t follows if V_t meets the open
where the big semiample contraction is an isomorphism. No such
positivity is asserted for every exceptional subvariety.
@@ -4080,18 +4069,18 @@ positivity is asserted for every exceptional subvariety.
Proof.
On the smooth marked-point open, evaluation into the finite jet
bundle is a morphism from the constant bundle
-$H^0(Z,kP) ⊗ cO$ to a locally free sheaf. Remove the rank-jump
+H^0(Z,kP) ⊗ cO to a locally free sheaf. Remove the rank-jump
loci for the finitely many levels. Its kernels are then vector
subbundles whose fibers are exactly the full spaces
-$mathcal H_i(x)$.calH_i(x)$. Their base loci are nested in increasing order
-as $i$ increases. Starting with the stipulated component through
+mathcal H_i(x).calH_i(x). Their base loci are nested in increasing order
+as i increases. Starting with the stipulated component through
the mark, follow a component containing it at each later level.
-Every such component has dimension between $1$ and $d-1$: the
-kernel is nonzero, so its base locus is proper. Among the $d+1$
+Every such component has dimension between 1 and d-1: the
+kernel is nonzero, so its base locus is proper. Among the d+1
nested components two successive ones have the same dimension
and hence are equal.
-This tracking can be made algebraic for the fixed $k$. Follow the
+This tracking can be made algebraic for the fixed k. Follow the
finitely many components of the geometric generic base loci,
their inclusions, and their incidences with the mark. They are
defined after a finite extension of the generic parameter field.
@@ -4099,67 +4088,67 @@ Spread over a corresponding irreducible parameter variety and
shrink for the required flatness, geometric integrality, and
constant-rank conditions. There are only finitely many choices
of adjacent step; one of them gives a family dominant over the
-marked-point open. It gives the asserted $T$ and $Gamma$.
-The evaluation $Gammato Z$ is dominant because the moving mark
-belongs to each $V_t$ and its map to $Z$ is dominant.
+marked-point open. It gives the asserted T and Gamma.
+The evaluation Gammato Z is dominant because the moving mark
+belongs to each V_t and its map to Z is dominant.
We now prove ordinary high-order vanishing. Work on a smooth
parameter open, and denote the higher-kernel bundle there by
-$mathcal K$.calK$. Evaluation on $Ttimes Z$ gives a map from the
-pullback of $mathcal K$calK$ to the pullback of $cO_Z(kP)$.
-Let $mathfrak b_T$frakb_T$ be its base ideal. After trivializing the
-target line bundle, a local frame of $mathcal K$calK$ generates
-$mathfrak b_T$.frakb_T$. We show that every frame section lies in
-$mathcal I_calI_Gamma^h$.
+mathcal K.calK. Evaluation on Ttimes Z gives a map from the
+pullback of mathcal KcalK to the pullback of cO_Z(kP).
+Let mathfrak b_Tfrakb_T be its base ideal. After trivializing the
+target line bundle, a local frame of mathcal KcalK generates
+mathfrak b_T.frakb_T. We show that every frame section lies in
+mathcal I_calI_Gamma^h.
Each frame element has the form
[
s(t,z)=sum_alpha c_alpha(t)s_alpha(z),
]
-where the $s_alpha$ are a fixed basis of the global space
-$H^0(Z,kP)$. Parameter differentiation with $z$ fixed only
+where the s_alpha are a fixed basis of the global space
+H^0(Z,kP). Parameter differentiation with z fixed only
differentiates the coefficients. Every derivative, at each fixed
parameter, is consequently a global section of the same line
-bundle $kP$.
+bundle kP.
-Write $x(t)$ for the moving mark. In local coordinates $z-x(t)$,
-the section starts in order $lceil ktau_{i+1}rceil$. Each parameter
+Write x(t) for the moving mark. In local coordinates z-x(t),
+the section starts in order lceil ktau_{i+1}rceil. Each parameter
derivative lowers this order by at most one. A derivative of
-order $r$ therefore belongs to the lower kernel if
+order r therefore belongs to the lower kernel if
[
rleqlceil ktau_{i+1}rceil-lceil ktau_irceil.
]
-The right side is at least $kdelta-1$. Since $kdeltageq4$,
-every derivative of order less than $h=lceil kdelta/2rceil$ is
-allowed. It thus vanishes along $Gamma$, which lies in the
+The right side is at least kdelta-1. Since kdeltageq4,
+every derivative of order less than h=lceil kdelta/2rceil is
+allowed. It thus vanishes along Gamma, which lies in the
lower base locus. A change of higher-kernel frame contributes
only parameter coefficients and derivatives of orders no larger
-than $r$, so the conclusion holds for any local frame.
+than r, so the conclusion holds for any local frame.
Here a first-order normal-space assertion must be upgraded to
-all orders. At a general smooth point of $Gamma$, its projection
-to $Z$ is submersive. Thus
+all orders. At a general smooth point of Gamma, its projection
+to Z is submersive. Thus
[
T_{(t,z)}(Ttimes Z)
=T_{(t,z)}Gamma+(T_tTtimes{0}).
]
In particular, parameter directions span the normal space to
-$Gamma$. Choose parameter coordinates
-$t=(t',t'')$, where $t'$ has length
-$a=codim_{Ttimes Z}Gamma$, such that the normal equations have
-an invertible Jacobian minor in the $t'$ directions. The analytic
+Gamma. Choose parameter coordinates
+t=(t',t''), where t' has length
+a=codim_{Ttimes Z}Gamma, such that the normal equations have
+an invertible Jacobian minor in the t' directions. The analytic
implicit-function theorem writes
[
- Gamma:quad t'=F(t'',z).
+ Gamma: t'=F(t'',z).
]
-Set $u=t'-F(t'',z)$. With $t''$ and $z$ held fixed, differentiation
-in $t'$ is exactly differentiation in $u$, to every order. Use a
-frame of $kP$ depending only on $z$. The vanishing on $Gamma$
-of all parameter derivatives of order below $h$ says that every
-normal Taylor coefficient of degree below $h$ vanishes. Therefore
-$sinmathcal I_calI_Gamma^h$ in the ordinary analytic local ring.
+Set u=t'-F(t'',z). With t'' and z held fixed, differentiation
+in t' is exactly differentiation in u, to every order. Use a
+frame of kP depending only on z. The vanishing on Gamma
+of all parameter derivatives of order below h says that every
+normal Taylor coefficient of degree below h vanishes. Therefore
+sinmathcal I_calI_Gamma^h in the ordinary analytic local ring.
Equivalently, in arbitrary relative normal coordinates a product
-of $r$ normal vector fields expands into parameter derivatives
-of orders at most $r$, so no unaccounted higher-coordinate terms
+of r normal vector fields expands into parameter derivatives
+of orders at most r, so no unaccounted higher-coordinate terms
arise.
The sections and ideals are algebraic. Faithful flatness of the
@@ -4167,20 +4156,20 @@ analytic local ring over the algebraic local ring gives the same
ideal membership algebraically. Applying this to a finite frame
and clearing the finitely many denominators gives containment
on a dense algebraic open of the incidence. Generic smoothness
-of $Gammato T$ identifies its scheme fiber there with the
-reduced general component $V_t$. Restriction therefore gives
-$mathcal I_calI_Gamma^hcO_{{t}times Z}=mathcal I_calI_{V_t}^h$ locally.
+of Gammato T identifies its scheme fiber there with the
+reduced general component V_t. Restriction therefore gives
+mathcal I_calI_Gamma^hcO_{{t}times Z}=mathcal I_calI_{V_t}^h locally.
The specialized frame is a basis of the entire higher kernel,
so the containment concerns its whole base ideal.
-If $Z$ is $Q$-factorial and klt, apply
+If Z is Q-factorial and klt, apply
Lemma~common:base-component to the higher kernel.
-Its generic base ideal is primary because $V_t$ is a base-locus
+Its generic base ideal is primary because V_t is a base-locus
component, and its generic point is smooth by incidence dominance.
-The bounds follow from $k/hle2/delta$. On the isomorphism open
-of the big semiample contraction the image of $V_t$ has dimension
-$f$, so the pullback of an ample class has positive top intersection
-on $V_t$. This proves the stated positivity qualification.
+The bounds follow from k/hle2/delta. On the isomorphism open
+of the big semiample contraction the image of V_t has dimension
+f, so the pullback of an ample class has positive top intersection
+on V_t. This proves the stated positivity qualification.
## Restriction to a suitable fiber
@@ -4190,20 +4179,20 @@ global positivity and singularity assumptions of subadjunction.
[Restriction of an isolated base ideal]
-In the setting of Lemma~common:moving-kernel, let $g:Zto B$
+In the setting of Lemma~common:moving-kernel, let g:Zto B
be a morphism. There is a dense open of the incidence on which
-the following holds. Choose $(t,z)$ in that open, let $b=g(z)$,
-and suppose that $V_t$ is smooth over the smooth open of its
-actual image at $z$. Let $F$ be the reduced component through
-$z$ of its scheme fiber. Locally on $Z_b$ at $z$, restriction of
-the higher kernel's base ideal has support exactly $F$ and is
-contained in $mathcal I_F^h$.calI_F^h$.
-If $F$ is proper in an integral ambient fiber component containing
-it, the restricted series is nonzero there and $F$ is an isolated
+the following holds. Choose (t,z) in that open, let b=g(z),
+and suppose that V_t is smooth over the smooth open of its
+actual image at z. Let F be the reduced component through
+z of its scheme fiber. Locally on Z_b at z, restriction of
+the higher kernel's base ideal has support exactly F and is
+contained in mathcal I_F^h.calI_F^h.
+If F is proper in an integral ambient fiber component containing
+it, the restricted series is nonzero there and F is an isolated
base component at its generic point. Application of
Lemma~common:base-component on that ambient component
-requires all its hypotheses, including projectivity, $Q$-factoriality,
-klt singularities, and smoothness at the generic point of $F$,
+requires all its hypotheses, including projectivity, Q-factoriality,
+klt singularities, and smoothness at the generic point of F,
separately; they are not conclusions of this restriction lemma.
@@ -4214,20 +4203,20 @@ failures of generic smoothness for the incidence over its parameter
space and for the component over its actual image. These are proper
closed subsets after shrinking. A prescribed very general condition
on the evaluation can also be imposed because the incidence dominates
-$Z$. The component is then chosen through this point, rather than
+Z. The component is then chosen through this point, rather than
by selecting a possibly special fiber in advance.
-Locally the original base support is exactly $V_t$. Smoothness
+Locally the original base support is exactly V_t. Smoothness
over the actual image makes its scheme fiber reduced at the chosen
point. Thus in the ambient fiber
[
- mathcal I_calI_{V_t}mathcal O_calO_{Z_b}=mathcal I_F,calI_F,qquad
+ mathcal I_calI_{V_t}mathcal O_calO_{Z_b}=mathcal I_F,calI_F,
mathcal I_calI_{V_t}^{h}mathcal O_calO_{Z_b}=mathcal I_F^calI_F^{h}
]
near that point. Restricting the actual sections extends their
base ideal, which proves both asserted local properties.
No other base component contains the chosen point, so none can
-contain the component $F$ through it. If this component is proper
+contain the component F through it. If this component is proper
in the chosen integral ambient component, the base support is
proper there, and the restricted series cannot vanish identically.
At its generic point the resulting ideal is primary. This is a
@@ -4247,27 +4236,27 @@ the base into finitely many normal covers.
[Finite covers after excluding sweeping branch divisors]
-Let $Y$ be a normal projective complex variety. Let
-$mathcal VcalVto T$ be a smooth projective family with integral
+Let Y be a normal projective complex variety. Let
+mathcal VcalVto T be a smooth projective family with integral
fibers, over an integral parameter variety, and let
-$g:mathcal VcalVto Y$ be a morphism whose general fiber maps
-$g_t:V_tto Y$ are dominant and generically finite of degree
-at most a fixed integer $e$.
+g:mathcal VcalVto Y be a morphism whose general fiber maps
+g_t: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 $Y_{rm sm}$ to divisors
-$mathcal R_1,...,calR_1,...,mathcal R_N$calR_N$ in $mathcal V$.calV$.
-Retain the notation $mathcal V,T,g$calV,T,g$ after this base change.
-Let $I$ be the set of indices $j$ for which the image closure
-$g_t((mathcal R_j)_t)bar$calR_j)_t)bar$ is a divisor in $Y$ for general $t$.
+Jacobian over Y_{rm sm} to divisors
+mathcal R_1,...,calR_1,...,mathcal R_NcalR_N in mathcal V.calV.
+Retain the notation mathcal V,T,gcalV,T,g after this base change.
+Let I be the set of indices j for which the image closure
+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).
]
Then, after shrinking again, one smooth nonempty open
-$Y^circsubset Y$ makes every general finite normal Stein cover
-of $g_t$ finite 'etale over $Y^circ$. There are only finitely
-many such covers of $Y$, up to isomorphism over $Y$.
+Y^circsubset Y makes every general finite normal Stein cover
+of g_t finite 'etale over Y^circ. There are only finitely
+many such covers of Y, up to isomorphism over Y.
The same conclusion holds simultaneously for finitely many
projections. The open and finite lists may depend on all fixed
parameters of the family.
@@ -4279,14 +4268,14 @@ the general maps are generically finite in characteristic zero.
Its zero divisor has finitely many irreducible components.
Shrink away vertical components and make a finite parameter extension
to define the geometric generic components individually. Their
-closures give the divisors $mathcal R_j$calR_j$ in the statement.
+closures give the divisors mathcal R_jcalR_j in the statement.
Shrink further to keep their fiber-image dimensions constant.
These components account for every branch prime of the general
finite Stein cover: a ramification prime there has a strict
-transform on $V_t$, because a proper birational morphism to a
+transform on V_t, because a proper birational morphism to a
normal variety is an isomorphism at each generic divisorial
point. A divisor exceptional over the Stein cover has image of
-codimension at least two on that cover and hence on $Y$, so it
+codimension at least two on that cover and hence on Y, so it
cannot hide a branch prime. Components confined to special
parameters disappear upon shrinking.
@@ -4295,26 +4284,26 @@ Put
B=Y_{rm sing}cup∪_{jin I}g(mathcal R_j)bar.calR_j)bar.
]
Each closed set in this finite union is proper by hypothesis,
-so $Bne Y$. The finite normal covers are unramified in
-codimension one over the smooth open $Y^circ=Y∖ B$.
+so Bne Y. The finite normal covers are unramified in
+codimension one over the smooth open Y^circ=Y∖ B.
Purity of the branch locus makes them finite 'etale there
cite[Expos'e~X, Theorem~3.1]{SGA1}.
-The topological fundamental group of $Y^circ$ is finitely
+The topological fundamental group of Y^circ is finitely
generated. Indeed, triangulate the compact semialgebraic space
-$Y$ compatibly with its closed subset $B$
+Y compatibly with its closed subset B
cite[Theorem~1.10 in the author's notes]{Coste2007}.
-After barycentric subdivision the subcomplex for $B$ is full.
+After barycentric subdivision the subcomplex for B is full.
On its complement, normalize the sum of the barycentric
coordinates at vertices outside that subcomplex; decreasing
the other coordinates to zero gives a deformation retraction
onto a finite subcomplex. Its fundamental group is finitely
generated. A finitely generated group has only
finitely many permutation representations in each symmetric
-group of degree at most $e$. Riemann existence
+group of degree at most e. Riemann existence
cite[Expos'e~XII, Theorem~5.1]{SGA1} therefore gives
finitely many finite 'etale covers of these degrees. A finite
-normal cover of $Y$ is the normalization in the function field
-of its restriction to $Y^circ$, and is determined by that
+normal cover of Y is the normalization in the function field
+of its restriction to Y^circ, and is determined by that
restriction. Taking the union of the proper image closures for
finitely many projections proves the simultaneous assertion.
@@ -4356,8 +4345,8 @@ An integer (q>0) will be chosen after (ε) and before (t).
As rational (t↓0),
choose positive integers (r=r(t)) with
- rmu_t→ε,qquad
- L=r(K+tA),qquad L_b=L+bmKquad(0le ble q).
+ rmu_t→ε,
+ L=r(K+tA), L_b=L+bmKquad(0le ble q).
Only rational weights (b) are used for model constructions;
volume functions in integrals are extended continuously to real weights.
@@ -4365,7 +4354,7 @@ The positive part of (L_b) is
the nef semiample class
[
N_b=(r+bm)(K_s+sA_s)
- quadhbox{on }Y_s,qquad s=(rt)/(r+bm).
+ quadhbox{on }Y_s, s=(rt)/(r+bm).
]
When emphasizing the weight, denote this axis model by (Y(b)=Y_s)
and write (A_b) for its transform of (A).
@@ -4384,7 +4373,7 @@ More explicitly, for every fixed (q), once (t) is sufficiently
small depending on (q),
(ε^n)/(2)le N_b^nle2ε^n
- qquad(0le ble q).
+ (0le ble q).
The constants in this display are independent of (q); only the
threshold for (t) depends on the fixed choice of (q).
@@ -4412,13 +4401,13 @@ of jets in one Cartier degree (kP), at levels separated by a gap
(delta). It gives vanishing along (V) in the ordinary ideal power
of order
[
- h=lceil kdelta/2rceil,qquad kdeltage4,
+ h=lceil kdelta/2rceil, kdeltage4,
]
and, when (Z) is (Q)-factorial and klt, the estimates
P^fcdot Vle (2/delta)^{d-f}P^d,
- qquad K_{V^*}P^{f-1}le(K_Z+cP)|_V P^{f-1},
- quad 0<cle 2(d-f)/deltale2d/delta.
+ K_{V^*}P^{f-1}le(K_Z+cP)|_V P^{f-1},
+ 0<cle 2(d-f)/deltale2d/delta.
Here (V^*) is a smooth resolution of (V). The canonical estimate
uses Lemma~common:numerical-subadjunction, which allows the nef
@@ -4426,7 +4415,7 @@ class on the center itself and requires log canonicity only near its
generic point. The last, weaker coefficient is sufficient here. At every
application we choose the incidence point in the isomorphism open of the
big semiample contraction. Dominance of the incidence permits this choice;
-it ensures that $P|_V$ is big and hence $P^fcdot V>0$.
+it ensures that P|_V is big and hence P^fcdot V>0.
Exceptional components not meeting that open are not assigned this
positivity. The separate fiber-restriction Lemma~common:fiber-restriction
will be used only at a general incidence point where the scheme fiber is
@@ -4437,7 +4426,7 @@ genus as well as bounded degree. The preceding intersection estimates
provide them on components of general type. Suppose a moving
(f)-fold (V) is of general type, (P|_V) is nef and big, and
- 0<P^f.Vle C_0,qquad
+ 0<P^f.Vle C_0,
K_{V^*}P^{f-1}le C_1P^f.V.
Effective birationality gives a uniform pluricanonical degree whose
@@ -4537,8 +4526,8 @@ normalized adjoint divisor on a projective bundle. The weight
decomposition of its sections will let us compare moving components
with either copy of the base. On (Ytimes Y), let
- Z_0=PP(mK_1 ⊕ mK_2),qquad
- xi=OO_{Z_0}(1),qquad M=L_1+L_2+qxi.
+ Z_0=PP(mK_1 ⊕ mK_2),
+ xi=OO_{Z_0}(1), M=L_1+L_2+qxi.
We use the convention that sections of (kxi) are symmetric powers
of the indicated direct sum. Fixing one base slot gives the
@@ -4551,7 +4540,7 @@ The torus open in either bundle is the complement of its two axes.
Both the total class (M) and the slice class have big semiample
positive parts (P) on (Q)-factorial klt models (Z), with
[
- K_Z+Deltasim_{Q}c_tP,qquad Deltage0,
+ K_Z+Deltasim_{Q}c_tP, Deltage0,
]
where (c_t) is bounded independently of small (t).
Their volumes satisfy, respectively,
@@ -4564,7 +4553,7 @@ In particular both volumes are bounded above and below by positive
constants times (q). For each rational weight, on common
resolutions,
- Psim_{Q}N_{b,1}+N_{q-b,2}+E_b,qquad E_bge0,
+ Psim_{Q}N_{b,1}+N_{q-b,2}+E_b, E_bge0,
with the analogous one-term formula on slices.
At a generic point of a base divisor of a slice, and on a general
@@ -4589,14 +4578,14 @@ canonical singularities are canonical, and the projective bundles
are klt. The two disjoint Cartier axes form a plt boundary (D),
and
[
- K_{Z_0}+D=K_{rm sum},qquad
+ K_{Z_0}+D=K_{rm sum},
Dsim2xi-mK_{rm sum}.
]
The notation (K_{rm sum}) means (K_1+K_2), or (K) on a slice.
For (0<sigmale1), put
[
- gamma=2sigma+frac{q(1+sigma m)}r,qquad
+ gamma=2sigma+frac{q(1+sigma m)}r,
Xi_tsim_{Q}xi+
((1+sigma m)t)/(gamma)A_{rm sum}.
]
@@ -4670,7 +4659,7 @@ In the range of Equation~(jet:uniform-normalized-volume),
the bounds needed when choosing (q) are explicitly
[
vol(M)ge ((2n+1)!)/(4(n!)^2),qε^{2n},
- qquad
+
vol(L+qxi)le2(n+1)qε^n.
]
Their coefficients are independent of (q); the smallness threshold
@@ -4813,7 +4802,7 @@ the preceding restriction gives ordinary multiplicity at least
K_{F^*}P_{sl}^{f-1}
&le(K_{Z_{sl}}+cP_{sl})|_F
P_{sl}^{f-1},
- qquad 0<cle2(n+1-f)/delta.
+ 0<cle2(n+1-f)/delta.
]
@@ -4884,7 +4873,7 @@ dimension (c=n-w). Resolve the finitely many coefficient ideals
simultaneously. The exponents for which their product defines an lc
ideal pair form the rational polytope
[
- u_lge0,qquad
+ u_lge0,
sum_lu_lord_E(I_l)le a(E;Y,0).
]
Here (E) runs through the divisors needed to test log canonicity on
@@ -4918,7 +4907,7 @@ coordinate still has center in (V(I_l)).
Set
[
b=(sum_lu_l(l/k))/(sum_lu_l),
- qquad c'=ksum_lu_lle(kc)/(h).
+ c'=ksum_lu_lle(kc)/(h).
]
Since a weight-(l) coefficient divisor has class (kL_{l/k}),
the constructed boundary has class (c'L_b). Also
@@ -5047,7 +5036,7 @@ Terminal (Y(b)) is smooth in codimension
two. Thus normalization and conductor adjunction along (J)
give
[
- K_{J^nu}+C=(K_{Y(b)}+J)|_{J^nu},qquad Cge0
+ K_{J^nu}+C=(K_{Y(b)}+J)|_{J^nu}, Cge0
]
in codimension one. On resolving (J^nu), exceptional divisors
map to codimension at least two and pair trivially with the
@@ -5074,7 +5063,7 @@ It follows that, for all sufficiently small (t), all divisorial
branch components in the chosen family are fixed.
Fixed covers.:
-We now fix $t$. The branch complement is chosen from the one finite-type
+We now fix t. The branch complement is chosen from the one finite-type
family of correspondences under consideration, before any cover is
classified. After a finite parameter extension and shrinking, resolve
the two evaluation maps simultaneously over a smooth parameter open
@@ -5084,15 +5073,15 @@ excludes each component whose divisorial images sweep the base: such a
component would supply the bounded-genus covering curves of
Proposition~prop:bounded-curves. For every remaining component the
closure of its divisorial evaluation image is a fixed proper closed
-subset of $Y$, and the intersection estimates already bound the degrees
+subset of Y, and the intersection estimates already bound the degrees
of both projections. These are precisely the hypotheses of
Lemma~common:finite-covers for the smooth projective family of
-resolved correspondences. It gives a smooth dense open $Y^0$ and only
+resolved correspondences. It gives a smooth dense open Y^0 and only
finitely many finite normal Stein covers in either slot, all etale over
-$Y^0$. Bad parameter fibers are removed by shrinking the parameter space,
-not by deleting their possibly dominant images in $Y$. Both $Y^0$ and
-the finite cover lists may depend on this fixed $t$; no common complement
-or list as $tto0$ is used.
+Y^0. Bad parameter fibers are removed by shrinking the parameter space,
+not by deleting their possibly dominant images in Y. Both Y^0 and
+the finite cover lists may depend on this fixed t; no common complement
+or list as tto0 is used.
For a general member (V_a^*) of the resolved family, factor the two
projections through their finite normal Stein covers. The preceding
@@ -5149,51 +5138,51 @@ determinant factor, producing the opposite estimate.
For a vector bundle on a smooth projective curve, its slope is
degree divided by rank. The least and greatest Harder--Narasimhan
-slopes are denoted by $mu_{min}$ and $mu_{max}$.
+slopes are denoted by mu_{min} and mu_{max}.
For a line bundle on a smooth variety, to generate jets through
-order $a$ at a point means surjectivity to its stalk modulo
-the $(a+1)$-st power of the maximal ideal.
+order a at a point means surjectivity to its stalk modulo
+the (a+1)-st power of the maximal ideal.
[Finite-data Frobenius incompatibility]
-Let $W$ be a smooth connected projective complex variety of
-dimension $nge2$. Let $L,H$ 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 L,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}le2H^n/r,qquad
+ H^nle(2ε)^n,
+ K_WH^{n-1}le2H^n/r,
(2L+qS)H^{n-1}le4H^n,
- (14ε)^n<tfrac14,qquad 80ε<tfrac34.
+ (14ε)^n<tfrac14, 80ε<tfrac34.
Assume that the following fixed data exist.
[label=(roman*)]
-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 $l_i$, 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, with
+mu_{min}(Ω_W^1|_C)ge0.
+ On the projective bundle
[
Z=mathbf P(bfP(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=L_1+L_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. A smooth
-point $z_*$ in the complement of the two axes and finitely many
-sections of $k_0M$, for some integer $k_0>0$ with $k_0L$ Cartier,
-generate jets through order $(2n+2)k_0$ at $z_*$.
-item A point $xin W$, its blowup
-$pi_x:widehatwidehat WWto W$ with exceptional divisor $J$, and a
+ xi=c_1(mathcal O_Z(1)),calO_Z(1)), M=L_1+L_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. A smooth
+point z_* in the complement of the two axes and finitely many
+sections of k_0M, for some integer k_0>0 with k_0L Cartier,
+generate jets through order (2n+2)k_0 at z_*.
+ A point xin W, its blowup
+pi_x:widehatwidehat WWto W with exceptional divisor J, and a
curve class
[
gamma=f_*(A_1... A_{n-1}),
]
-where $f:Vtowidehatwidehat W$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 f: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^*(L+K_W/2+lambda S)cdotgamma
le40ε Jcdotgammaqquad(0lelambdale q).
@@ -5203,9 +5192,9 @@ It suffices to check the two endpoints in this affine inequality.
These data cannot coexist.
-There is no relation required between $x$ and the two base
-coordinates of $z_*$. The divisors $L,S,K_W$ are not assumed
-nef, and $K_W$ is not assumed pseudo-effective. The theorem
+There is no relation required between x and the two base
+coordinates of z_*. The divisors L,S,K_W are not assumed
+nef, and K_W is not assumed pseudo-effective. The theorem
starts with the actual flag and movable witness, rather than
with geometric conditions from which one might construct them.
All data in its statement are fixed before the residual
@@ -5214,17 +5203,17 @@ characteristic tends to infinity.
### Spreading the fixed witnesses
We prove the theorem in the next five steps. Put
-$Λ=prod_{i=1}^{n-1}l_i$. The flag satisfies
+Λ=prod_{i=1}^{n-1}l_i. The flag satisfies
- [C]=Λ H^{n-1},qquad h_icdot C=l_iΛ H^n.
+ [C]=Λ H^{n-1}, h_icdot C=l_iΛ H^n.
-Choose one sufficiently ample integral Cartier divisor $T_0$ so
-that every $T_0+aS$, $0le ale(k_0-1)q$, has a section
-nonvanishing at each of the two base coordinates of $z_*$.
+Choose one sufficiently ample integral Cartier divisor T_0 so
+that every T_0+aS, 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 $gamma$,
+points, the flag, the complete-intersection witness for gamma,
and the jet and filler sections spread over an integral finitely
-generated $mathbf Z$-algebrabfZ$-algebra of characteristic zero. Shrink its
+generated mathbf Z-algebrabfZ-algebra of characteristic zero. Shrink its
spectrum so that the fibers and flag are smooth projective and
geometrically integral, the birational morphisms remain birational,
the fixed ample bundles remain ample, and the finite jet
@@ -5232,13 +5221,13 @@ surjection and nonvanishings persist. Birationality is preserved
by spreading an isomorphism between dense opens. All displayed
intersection numbers are constant.
-We also preserve $mu_{min}(Ω_W^1|_C)ge0$.
+We also preserve mu_{min}(Ω_W^1|_C)ge0.
One way is to spread its characteristic-zero Harder--Narasimhan
filtration, make all its factors locally free, and use openness
of their semistability on curves; their 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 $d$.
+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 many ranks and
degrees. The proper relative Quot schemes for those Hilbert
polynomials have images missing the generic point: after removal
@@ -5249,66 +5238,66 @@ finitely many images proves the same openness assertion directly.
This base has closed points in arbitrarily large characteristics.
Take algebraic closures of their residue fields and exclude
denominator primes. Retain the notation for the reductions.
-The curve $gamma$ continues to pair nonnegatively with every
+The curve gamma continues to pair nonnegatively with every
effective divisor: pull that divisor back under the fixed
birational morphism and intersect with its fixed ample divisors.
This tests effective divisors on the reduction itself. No
specialization of a characteristic-zero effective cone is used.
-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_pL+T_0.
+ N_p=lfloor p/k_0rfloor, B_p=k_0N_pL+T_0.
]
-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 its local ring by the $p$-th powers of all regular parameters.
+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 its 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. This argument takes
place in the local ring and never divides by factorials.
-Put $d_p=(p-k_0N_p)q$, so $0le d_ple(k_0-1)q$.
+Put d_p=(p-k_0N_p)q, so 0le d_ple(k_0-1)q.
Among the sections fixed before reduction, choose
[
- u_0in H^0(W,cO_W(T_0)),qquad
+ u_0in H^0(W,cO_W(T_0)),
v_{d_p}in H^0(W,cO_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
-fiber monomial is the $d_p$-th power of the second coordinate.
+fiber 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 $z_*$, since $z_*$ lies
+d_pxi+T_{0,1}+T_{0,2} nonzero at z_*, since 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
+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.
+values of 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 comparison is
+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)]{MustataSchwede2014}.
### Full rank off the diagonal
-Let $F:Wto W'$ be 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 F:Wto W' be 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_*cO_W(B_p),quad s=rkmathcal E=p^n,calE=p^n,quad R=s^2,
- qquad U_a=H^0(W,cO_W(B_p+paS)).
+ mathcal E=F_*calE=F_*cO_W(B_p), s=rkmathcal E=p^n,calE=p^n, R=s^2,
+ U_a=H^0(W,cO_W(B_p+paS)).
]
-Projection formula gives an evaluation map on $W'times W'$:
+Projection formula gives an evaluation map on W'times W':
⊕_{substack{a+b=qa,bge0}}
U_a ⊗ U_b ⊗
@@ -5317,40 +5306,40 @@ Projection formula gives an evaluation map on $W'times W'$:
-The map in Equation~(common:two-slot-evaluation) has generic rank $R$.
+The map in Equation~(common: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
-Lemma~common:positive-characteristic-jets are weight polynomials in $z$
-with weight-$j$ coefficients 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~common:positive-characteristic-jets are weight polynomials 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.
+ H^0(W,B_p+(pq-j)S), 0le jle pq.
]
-Let $A_*$ be the tensor product of the local algebras of the two
+Let A_* be the tensor product of the local algebras of the two
base Frobenius fibers. The local quotient in
Lemma~common: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
Equation~(common: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 evaluation has full
+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 $A_*$,
+calculation uses a basis over the possibly nonreduced algebra A_*,
so reducedness of the Frobenius fibers is unnecessary.
-The jet calculation at $z_*$ proves generic full rank; the diagonal
-estimate below will be tested at $(x',x')$, obtained 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'), obtained from the
+separately fixed point x.
### The diagonal filtration
@@ -5361,59 +5350,59 @@ The Frobenius fiber product has the cartesian square
W arrow[r,"F"'] & W'.
-Its reduced subscheme is the diagonal $W$. Put
+Its reduced subscheme is the diagonal W. Put
[
mathcal L_p=calL_p=
cO_{Delta_F}(B_{p,1}+B_{p,2}+pqS_1).
]
-Write $h=Fcircpr_1=Fcircpr_2:
-Delta_Fto W'$. On this fiber product,
+Write h=Fcircpr_1=Fcircpr_2:
+Delta_Fto W'. On this fiber product,
[
cO(pS_1)simeq h^*cO_{W'}(S')simeqcO(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:
[
cO_{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
-Equation~(common:two-slot-evaluation) becomes $-qS'$.
+On the diagonal of W'times W', every source twist of
+Equation~(common:two-slot-evaluation) becomes -qS'.
Finite base change and 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 diagonal point
-therefore lie in the image of $H^0(Delta_F,mathcal L_p)$calL_p)$ in the
-corresponding fiber of $h_*mathcal L_p$.calL_p$. This proves that the rank
+of this one bundle mathcal L_p.calL_p. Their values at a diagonal point
+therefore lie in the image of H^0(Delta_F,mathcal L_p)calL_p) in the
+corresponding fiber of h_*mathcal L_p.calL_p. This proves that the rank
at every diagonal point is at most
h^0(Delta_F,mathcal L_p).calL_p).
-For a rank-$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
+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
[
- 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=cO_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
+parameter differences with their p-th powers zero, and its
+degree-one conormal is Ω_W^1. This coordinate calculation
can be made étale locally or in completions and identifies the
global associated graded. This is the diagonal-ideal form of the canonical Frobenius filtration
cite[Corollary~3.5]{KitadaiSumihiro2008}. The same calculation, using
-only the bundle from the second factor, filters $F^*mathcal E$calE$ with pieces
+only the bundle from the second factor, filters F^*mathcal EcalE with pieces
cite[Theorem~3.7]{Sun2008}
cO_W(B_p) ⊗ A_j(Ω_W^1).
@@ -5424,30 +5413,30 @@ Complementary multiplication is a perfect pairing, giving
A_{u-j}(V)simeq A_j(V)^vee ⊗ (det V)^{p-1}.
On monomials, each exponent vector pairs 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.
### One slope bound and one flag polynomial
We bound the sections of every graded bundle in
Equation~(common: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 absolute Frobenius of $C$. Langer's instability
+Write F_C for absolute Frobenius of C. Langer's instability
estimate cite[Corollary~2.5]{Langer2004} gives
L_{min}(V):=
@@ -5455,36 +5444,36 @@ estimate cite[Corollary~2.5]{Langer2004} gives
ge-(c)/(p-1).
To check the uniform constant, the discrepancy from
-$mu_{min}(V)ge0$ is at most
-$(rk V-1)deg A_C/(p-1)$, where $A_C$ is a nef line bundle such
-that $T_C ⊗ A_C$ is globally generated. Its degree can be
+mu_{min}(V)ge0 is at most
+(rk V-1)deg A_C/(p-1), where A_C is a nef line bundle such
+that T_C ⊗ A_C is globally generated. Its degree can be
bounded in terms of the fixed genus, using a sufficiently positive
-line bundle. Here the rank is $n$, irrespective of the fact that the base
+line bundle. Here the rank is n, irrespective of the fact that the base
is a curve.
-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
-Equation~(common:tensor-min). This limit is taken for each fixed $p,i$;
+Divide by p^e and let eto∞ to prove
+Equation~(common: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 Equation~(common:socle-pairing) yields
mu_{max}(A_j(V))
@@ -5494,41 +5483,41 @@ pairing Equation~(common:socle-pairing) yields
-The divisors $B_p-pL=T_0-(p-k_0N_p)L$ range over a fixed finite
+The divisors B_p-pL=T_0-(p-k_0N_p)L range over a fixed finite
list. Combining Equation~(common:intersection-bounds), Equation~(common:flag-degrees), Lemma~common: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 fixed. The coefficient $7$ provides harmless
+where c_0ge0 is fixed. 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_i h_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)
@@ -5536,7 +5525,7 @@ For each $b_kge0$, the divisor restriction sequence is
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
@@ -5544,14 +5533,14 @@ member of the flag bounds the section space by
h^0(C,mathcal G_j|_C(-calG_j|_C(-sum_i b_i h_i|_C)).
At each stage the remainder is zero after sufficiently negative
-ample twisting. Its stopping index need not be uniform in $j,p$
+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 Equation~(common: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 injective;
+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 Equation~(common:flag-degrees) it follows that
@@ -5563,91 +5552,91 @@ points. By Equation~(common:flag-degrees) it follows that
The leading coefficient follows from
-$Λ=prod_i l_i$. More importantly, the whole scalar
-polynomial on the first line is the same for every $j$.
+Λ=prod_i l_i. More importantly, the whole scalar
+polynomial on the first line is the same for every j.
Sum over the filtration in Equation~(common:diagonal-graded). The ranks
-satisfy $sum_j e_j=p^n$, so Equation~(common: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 Equation~(common: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 Equation~(common:intersection-bounds), proving the strict inequality
-for all large $p$.
+for all large p.
### The determinant has incompatible orders
-Choose $R$ generically independent columns of
+Choose R generically independent columns of
Equation~(common: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$, and the
+let m_i be the sum of the weights of the R chosen columns in
+slot i. The target determinant is
+(detmathcal E)^scalE)^sboxtimes(detmathcal E)^s,calE)^s, and the
chosen source determinant is
-$cO(-m_1S')boxtimescO(-m_2S')$. Thus
+cO(-m_1S')boxtimescO(-m_2S'). Thus
[
mathcal Q_i=(calQ_i=(detmathcal E)^calE)^{ ⊗ s}
- ⊗ cO_{W'}(m_iS'),qquad
+ ⊗ cO_{W'}(m_iS'),
lambda_i=m_i/R.
]
-Every column weight lies between $0$ and $q$, so
-$0lelambda_ile q$ independently of the columns chosen.
+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,
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]{Sun2008}; it is an identity of rational divisor classes. For the second equality, use
Equation~(common:Frob-pullback-graded). Complementary degrees in
Equation~(common:socle-pairing) have total first Chern class
-$e_j(p-1)K_W$. Summing gives
+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 gives
+For 0lelambdale q, the fixed movable inequality gives
[
(L+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
Equation~(common:slot-determinant-class) differs from its
-left-hand divisor at $lambda=lambda_i$ by
+left-hand divisor at lambda=lambda_i by
[
frac1p(B_p-pL-K_W/2).
]
Its numerator belongs to the fixed finite list
-$T_0-aL-K_W/2$, $0le a<k_0$. The resulting error after
-intersection with $gamma$ is therefore $O(p^{-1})$, uniformly
+T_0-aL-K_W/2, 0le a<k_0. 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 its effective divisor 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 its effective divisor 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 degree their
+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 their
leading terms are linearly independent. Tensor products of these
leading terms remain independent in each bidegree of the two
disjoint sets of parameters; terms of different bidegrees cannot
@@ -5659,25 +5648,25 @@ Equation~(common:slot-order), we obtain
On the other hand, Equation~(common:diagonal-rank-upper), Lemma~common: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
+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 the maximal ideal, so
-every term of the determinant has order at least $3R/4$. Therefore
+every term of the determinant has order at least 3R/4. Therefore
ord_{(x',x')}sigmage3R/4.
Equations (common:determinant-order-upper) and
(common:determinant-order-lower) contradict
-$80ε<3/4$ for sufficiently large $p$.
+80ε<3/4 for sufficiently large p.
This contradiction proves Theorem~common:finite-data-frobenius.
-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.
+not depend on p, the filtration index, or the chosen columns.
## The geometric witnesses and smooth nonvanishing
@@ -5706,7 +5695,7 @@ contradiction. We use the late terminal model (Y), the divisors
(K=K_Y) and (A=A_Y), and the scaling models of the preceding
section. Fix (m>0) with (mK) Cartier, and write
[
- mu_t=vol(K_X+tA_X)^{1/n},qquad
+ mu_t=vol(K_X+tA_X)^{1/n},
L=r(K+tA).
]
As before, (r) is a positive integer chosen so that
@@ -5716,7 +5705,7 @@ Propositions~prop:scalar-jets and~prop:large-seshadri, and also
so that
- (14ε)^n<frac14,qquad
+ (14ε)^n<frac14,
80ε<frac34.
The strict inequalities leave room for the errors that tend to zero
@@ -5737,15 +5726,15 @@ Choose a smooth common resolution (W) of (Y) and the scaling model
for (K+tA). In this application (K,A,L) also denote their pullbacks to
(W), and
[
- K_W=K+E,qquad Egeq0.
+ K_W=K+E, Egeq0.
]
Let (H_0) be the pullback of the nef semiample positive part (N_0) of
(L) on its scaling model. The comparison of canonical pullbacks and
moving parts gives
- H_0^n→ε^n,qquad
- LH_0^{n-1}=H_0^n,qquad
+ H_0^n→ε^n,
+ LH_0^{n-1}=H_0^n,
K_WH_0^{n-1}=KH_0^{n-1}leq(H_0^n)/(r).
To justify these intersections, recall that the maps
@@ -5768,8 +5757,8 @@ with (K_WH_0^{n-1}le H_0^n/r), allow us to choose (eta)
small enough that continuity gives
- H^nleq(2ε)^n,qquad
- K_WH^{n-1}leq(2H^n)/(r),qquad
+ H^nleq(2ε)^n,
+ K_WH^{n-1}leq(2H^n)/(r),
(2L+qmK)H^{n-1}leq4H^n.
@@ -5780,11 +5769,11 @@ cite[Theorem~2.1]{CampanaPaun2015}, and restriction to a sufficiently
general complete-intersection curve give the following fixed data.
Apply restriction to the Harder--Narasimhan filtration and its factors
cite[Theorem~6.1 and Remark~6.2]{MehtaRamanathan1982}. Choose successively
-large divisible integers $l_i$ such that $h_i=l_iH$ are integral very ample
+large divisible integers l_i such that h_i=l_iH are integral very ample
classes, and choose a smooth integral complete-intersection flag
[
W=W_nsupset W_{n-1}supset...supset W_1=C,
- qquad W_{i-1}in |h_{n-i+1}|_{W_i},
+ W_{i-1}in |h_{n-i+1}|_{W_i},
]
for which
@@ -5825,7 +5814,7 @@ actual covering curve class (gamma) on (widehatwidehat WW) such that
(pi_x^*D^+)cdotgamma
- <40ε,Jcdotgamma,qquad Jcdotgamma>0.
+ <40ε,Jcdotgamma, Jcdotgamma>0.
The strictness lets us fix an algebraic family that will survive
reduction. More precisely, the negative pairing can first be detected
@@ -5838,27 +5827,27 @@ rather than a limiting real movable class. Positivity of
pseudo-effectivity of (pi_x^*D^+).
To turn this curve into the witness required by
-Theorem~common:finite-data-frobenius, set $S=mK$ on $W$;
+Theorem~common:finite-data-frobenius, set S=mK on W;
it is an integral Cartier divisor. We verify all its weight inequalities
before proceeding to the jet construction. Put
-$Q_lambda=L+K_W/2+lambda S$ for $0lelambdale q$.
-The equality $K_W=K+E$ gives
+Q_lambda=L+K_W/2+lambda S for 0lelambdale q.
+The equality K_W=K+E gives
D^+-Q_lambda
=(r-tfrac12-lambda m)K+3rtA+tfrac32E.
-This class is pseudo-effective in characteristic zero: $r>qm+1$,
-$K$ is pseudo-effective, and $A,E$ are effective up to the stated
+This class is pseudo-effective in characteristic zero: r>qm+1,
+K is pseudo-effective, and A,E are effective up to the stated
equivalences. Pairing with the fixed strongly movable class therefore gives
[
pi_x^*Q_lambdacdotgamma
le pi_x^*D^+cdotgamma
<40ε Jcdotgamma,
- qquad Jcdotgamma>0.
+ Jcdotgamma>0.
]
These are the required movable inequalities. Only the two endpoint
inequalities need to be retained as finite data, since the pairing is
-affine in $lambda$. The common theorem spreads these fixed numerical
+affine in lambda. The common theorem spreads these fixed numerical
inequalities and the actual birational complete-intersection witness;
it does not spread the pseudo-effective assertion used to obtain them.
@@ -5870,8 +5859,8 @@ orders needed for arbitrarily large characteristics.
On (Wtimes W), let
[
- Z=PP(mK_1 ⊕ mK_2),qquad
- xi=OO_Z(1),qquad
+ Z=PP(mK_1 ⊕ mK_2),
+ xi=OO_Z(1),
M=L_1+L_2+qxi,
]
with the symmetric-power convention for the projective bundle.
@@ -5894,15 +5883,15 @@ and fix finitely many sections realizing this jet surjection.
### Applying the comparison
-All the preceding choices are now fixed. With $S=mK$, the dimension,
+All the preceding choices are now fixed. With S=mK, the dimension,
intersection and flag hypotheses of
Theorem~common:finite-data-frobenius are
Equations~(fr:epsilon), (fr:intersection-bounds),
and~(fr:generic-nef). The curve construction gives its movable
witness and both endpoint inequalities, and the finite jet system
-gives the required surjection at $z_*$. This point need not lie over
-the separately chosen testing point $x$. The proof of the common
-theorem also fixes its auxiliary divisor $T_0$ and finitely many
+gives the required surjection at z_*. This point need not lie over
+the separately chosen testing point x. The proof of the common
+theorem also fixes its auxiliary divisor T_0 and finitely many
filler sections before choosing any residual characteristic.
The common theorem now rules out these fixed data. Its contradiction
@@ -6053,7 +6042,7 @@ resolution (p:Wto X), (q:Wto V), compatible canonical divisors
satisfy
[
p^*K_X=q^*K_V+E,
- qquad Egeq0quadand E is q-exceptional.
+ Egeq0quadand E is q-exceptional.
]
In particular (kappa(X,K_X)=kappa(V,K_V)).
@@ -6083,7 +6072,7 @@ Proof.
Choose a positive integer (m) such that the line bundle
(L=OO_Z(mA)) is globally generated. Its complete system gives
[
- phi:Z→PP(H^0(Z,L)^*),qquad
+ phi:Z→PP(H^0(Z,L)^*),
Lsimeqphi^*OO(1).
]
The image has dimension at most (kappa(Z,A)=0), so it is a point.
@@ -6153,7 +6142,7 @@ Assume that (K_X+Delta) is nef and that, for an effective rational
divisor (D),
[
K_X+Deltasim_{Q}Dgeq0,
- qquad S⊆Supp D.
+ S⊆Supp D.
]
Then, for every integer (mgeq2) for which (m(K_X+Delta)) is Cartier,
the restriction map
@@ -6211,7 +6200,7 @@ give the usual multiplication on the section algebras below.
leavevmode
[label=(roman*)]
-item Let (k) be an algebraically closed field of characteristic zero,
+ Let (k) be an algebraically closed field of characteristic zero,
let ((X,Delta)) be a normal projective log canonical pair over (k), and assume
that (Deltageq0) is rational and (D=K_X+Delta) is (Q)-Cartier.
Then
@@ -6220,8 +6209,7 @@ Then
H^0(X,OO_X(lfloor mDrfloor))
]
is a finitely generated (k)-algebra.
-
-item Let ((X,Delta)) be a log canonical pair over (C), with
+ Let ((X,Delta)) be a log canonical pair over (C), with
(Deltageq0) rational and (K_X+Delta) (Q)-Cartier. For every
proper morphism (f:Xto S) onto an algebraic variety (S), the relative
log canonical algebra
@@ -6261,7 +6249,7 @@ Theorem~thm:main makes the rational adjoint (D_Y) semiample.
On a common smooth resolution (p:Wto X), (q:Wto Y), choose
compatible canonical divisors. Then
- p^*D=q^*D_Y+E,qquad Egeq0,qquad E is q-exceptional.
+ p^*D=q^*D_Y+E, Egeq0, E is q-exceptional.
Indeed, the coefficient of (E) at a prime divisor (F) is
(a(F;Y,Delta_Y)-a(F;X,Delta)), which is nonnegative by the discrepancy
@@ -6286,7 +6274,7 @@ the common function field,
[
H^0(X,OO_X(lfloor mDrfloor))
=H^0(Y,OO_Y(lfloor mD_Yrfloor))
- qquad(mgeq0).
+ (mgeq0).
]
These identifications preserve multiplication, so they identify the full
graded rings.
@@ -6353,8 +6341,8 @@ but not uniruled.
Let (X) be a smooth connected projective complex variety.
[label=(roman*)]
-item (X) is uniruled if and only if (kappa(X)=-∞).
-item (X) is rationally connected if and only if
+ (X) is uniruled if and only if (kappa(X)=-∞).
+ (X) is rationally connected if and only if
[
H^0(X,(Ω_X^1)^{ ⊗ m})=0
quadfor every integer m>0.
@@ -6397,13 +6385,13 @@ equivalence and is non-uniruled unless it is a point. We use
[Pluriforms and the rational quotient]
[label=(roman*)]
-item For every smooth connected projective complex variety (X),
+ For every smooth connected projective complex variety (X),
[
dim R(X)=
max_{substack{pinZ_{geq0},,kinZ_{>0}
H^0(X,Sym^k(Ω_X^p))ne0}} p.
]
-item A compact connected K"ahler manifold (X) is rationally
+ A compact connected K"ahler manifold (X) is rationally
connected if and only if
[
H^0(X,Sym^k(Ω_X^p))=0
@@ -6570,11 +6558,11 @@ in terms of bundles over abelian varieties.
Let (X) be a connected normal projective complex variety.
[label=(roman*)]
-item Its universal cover is biholomorphic to a quasi-projective variety
+ Its universal cover is biholomorphic to a quasi-projective variety
if and only if there is a finite 'etale Galois cover (X'to X)
and a locally trivial holomorphic fiber bundle (X'to A), where
(A) is an abelian variety and the fiber is simply connected and projective.
-item If (widetilde Xto X) is an infinite 'etale Galois cover
+ If (widetilde Xto X) is an infinite 'etale Galois cover
and (widetilde X) is biholomorphic to a quasi-projective variety,
there are a finite 'etale Galois cover (X'to X), a locally trivial
holomorphic fiber bundle (alpha:X'to A) over an abelian variety,
@@ -7082,7 +7070,7 @@ boundaries or semi-log-canonical analytic pairs.
% @article{Taji2014,
% author = {Taji, Behrouz},
-% title = {Birational positivity in dimension $4$},
+% title = {Birational positivity in dimension 4},
% journal = {Annales de l'Institut Fourier},
% year = {2014}, volume = {64}, number = {1}, pages = {203--216},
% doi = {10.5802/aif.2845},
@@ -7151,7 +7139,7 @@ boundaries or semi-log-canonical analytic pairs.
% @article{Miyaoka1988,
% author = {Miyaoka, Yoichi},
-% title = {Abundance conjecture for 3-folds: case {$nu=1$}},
+% title = {Abundance conjecture for 3-folds: case {nu=1}},
% journal = {Compositio Mathematica},
% volume = {68}, number = {2}, year = {1988}, pages = {203--220},
% url = {https://www.numdam.org/item/CM_1988__68_2_203_0/}
@@ -7241,7 +7229,7 @@ boundaries or semi-log-canonical analytic pairs.
% @misc{OpenAIMinimalModels2026,
% author = {{OpenAI}},
-% title = {{Minimal models and Mori fibre spaces for generalized log canonical $ℚ$-pairs}},
+% title = {{Minimal models and Mori fibre spaces for generalized log canonical ℚ-pairs}},
% howpublished = {OpenAI Math Release preprint
% OAI:Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026},
% year = {2026},