diff 56bc728..f6bf760 log_abundance_char_zero_v2.txt
diff --git a/log_abundance_char_zero_v2.txt b/log_abundance_char_zero_v2.txt
index 3b62f1a..58cf64e 100644
--- a/log_abundance_char_zero_v2.txt
+++ b/log_abundance_char_zero_v2.txt
@@ -769,7 +769,7 @@ 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
+klt, and P=floor(Delta) otherwise. Fix m>0 with mL
Cartier. For every rational
[
0<ε<(1)/(4nm+2),
@@ -785,9 +785,9 @@ 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.
+ -(L-tfrac12P)· Cle2n.
]
-Writing a=Lcdot Cge0 and p=Pcdot C, we have
+Writing a=L· Cge0 and p=P· C, we have
[
ple2a+4n, a<ε p, (1-2ε)a<4nε.
]
@@ -863,7 +863,7 @@ 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.
+Put P=Delta in the klt case and P=floor(Delta) 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
@@ -946,7 +946,7 @@ 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, b_i>0,
- qquadSupp G_V=D_V.
+ Supp 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
@@ -967,7 +967,7 @@ 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
[
- G_Vcdot H_V^{n-1}=(K_V+D_V)cdot H_V^{n-1}=0.
+ G_V· H_V^{n-1}=(K_V+D_V)· H_V^{n-1}=0.
]
Effectivity forces G_V=0, and its full support then gives D_V=0.
@@ -991,7 +991,7 @@ perturbed boundaries effective and klt. Convexity of the
pseudo-effective cone, applied between L and L-delta P, makes
both
[
- L-ε P, L-2ε P, P=lfloorDeltarfloor,
+ L-ε P, L-2ε P, P=floor(Delta),
]
pseudo-effective klt adjoints. Their Iitaka dimensions are zero:
nonvanishing gives the lower bound, and adding the effective
@@ -1008,7 +1008,7 @@ monotonicity and homogeneity of kappa_sigma give
]
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
+Lsim_{Q}Ege0. For an ample divisor H, E· H^{n-1}=0 forces
E=0. Hence Lsim_{Q}0, without any further nonvanishing or
abundance premise.
@@ -1039,7 +1039,7 @@ individual lemmas are for.
The semiample L|_D gives a morphism
[
phi:D→ P:=PP^s,
- N_0|_Dsimeqphi^*OO_P(1)
+ N_0|_D~=phi^*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
@@ -1051,13 +1051,13 @@ 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 XcalX with a reduced Cartier divisor
-Ssubsetmathcal XcalX and a finite representable morphism
-Sto Dtimes_Pmathcal PcalP whose image covers the positive boundary,
+a projective variety X with a reduced Cartier divisor
+SsubsetX and a finite representable morphism
+Sto Dtimes_PP 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
+ Sxrightarrow{g}Y{p}PP^s
]
with g finite representable and p a projective bundle. The
factorisation is the geometric output that the Hodge step of
@@ -1165,17 +1165,17 @@ 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
+X, 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*)]
-The pair (mathcal X,S+T)calX,S+T) is log canonical, and a fixed
+The pair (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).
+ omega_{X}+T)~=OO_{X}
tag{(ast_{adj})}
]
The support of T is locally set-theoretically principal.
@@ -1184,42 +1184,42 @@ Off T, the ambient space is Gorenstein and the displayed
isomorphism is ordinary Cartier adjunction data.
Put J=(Scap T)_{red}. Both S and J are Du Bois
-on scheme charts, the ideal mathcal I_calI_{J/S} is maximal
+on scheme charts, the ideal I_{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).
+ B:=calB:=
+ Hom_S( I_calI_{J/S},omega_S)
+ ~=OO_S(aS).
]
The identification agrees off J with the residue convention
specified by (sg:ambient-adjunction).
There is a finite representable morphism
[
- S→ Dtimes_Pmathcal PcalP
+ S→ Dtimes_PP
]
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.
+There is a smooth projective bundle p:Y PcalP
+and a representable projective morphism g:StoY.
This is a morphism from the reduced divisor S; no extension
-to a neighborhood of S in mathcal XcalX is asserted here.
+to a neighborhood of S in X 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,
+T={P}*OO_S, the morphism g factors
+through a closed embedding T↪calT↪Y,
and
[
- g_*OO_S=OO_{mathcal T}calT,
- OO_S(S)=g^*C, mathcal B=g^*C^a.calB=g^*C^a.
+ g_*OO_S=OO_{T},
+ OO_S(S)=g^*C, B=g^*C^a.calB=g^*C^a.
]
For a separated quasi-projective scheme chart
-Utomathcal YcalY that is etale and of finite type, put
-S_U=Stimes_{mathcal Y}calYU.
+UtoY that is etale and of finite type, put
+S_U=Stimes_{Y}
The etale morphism S_Uto S extends compatibly to every
-nilpotent thickening qS in mathcal X.calX.
+nilpotent thickening qS in X.
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
@@ -1236,11 +1236,11 @@ 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
+Let rho:R 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
+S_+,S_-,S_0 on R. 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
@@ -1248,48 +1248,48 @@ 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 R. With this full reduced
boundary, log ramification gives a log
canonical pair and the relation
[
- K_{mathcal R}calR+Bsim_{Q}a(S_+-S_-).
+ K_{R}+Bsim_{Q}a(S_+-S_-).
]
The ambient charts are klt. They are klt off the boundary,
and, on a log resolution, decreasing every boundary coefficient
slightly removes all zero-discrepancy places. The difference
between the two sides is a torsion integral Weil class, represented by
[
- mathcal F=calF=
- (omega_{mathcal R}calR(B)
- ⊗ OO_{mathcal R}calR(-a(S_+-S_-)))^{**}.
+ F=
+ (omega_{R}
+ ⊗ OO_{R}-a(S_+-S_-)))^{**}.
]
-Choose a periodicity isomorphism mathcal F^calF^{[q]}simeqOO_{mathcal R}calR,
+Choose a periodicity isomorphism F^{[q]}~=OO_{R},
where square brackets denote reflexive powers. It defines multiplication
in the finite algebra
[
- mathcal A=⊕_calA=⊕_{i=0}^{q-1}mathcal F^calF^{[i]}.
+ A=⊕_calA=⊕_{i=0}^{q-1}F^{[i]}.
]
-Let tau:mathcal R^calR^daggertomathcal RcalR be the normalization
-of Spec_{mathcal R}calRmathcal A.calA. In codimension one this is
+Let tau:R^ RcalR be the normalization
+of Spec_{R} 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.
+their pullbacks to R^
Tautological evaluation on this cover trivializes the pullback
-of mathcal FcalF in codimension one and hence fixes the adjoint
+of F 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
+Let p_1:Rtilde 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
- xrightarrow{tau}mathcal RcalRxrightarrow{rho}X.
+ Rtilde{p_1}R^
+ xrightarrow{tau}R{rho}X.
]
The blown-up ideal is
locally generated by two elements, so the ordinary blowup embeds
@@ -1306,7 +1306,7 @@ Cartier, and the divisors
[
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 Rtilde.
Since no codimension-two two-branch stratum is contained
in S_0, its pullback p_1^*S_0 is reduced Cartier and has no
exceptional component. The total boundary
@@ -1325,17 +1325,17 @@ is effective Cartier, ord_F(B')>0, whence
]
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,
+higher-codimension singular loci. On Rtilde,
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).
+ omega(S+T)~=OO(aS).
]
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.
+We also need finiteness of Sto S_+ on R^
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
@@ -1351,11 +1351,11 @@ 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 RtildecalRtilde the line OO(S-S'_-) has ell-th power equal to the pullback
+On Rtilde 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
+roots of N_0 on X. Denote this gerbe by G, and let
+kappa:Rtilde XcalX be the relative
+coarse-space morphism over G. 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,
@@ -1367,7 +1367,7 @@ 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
+now lie on X. 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.
@@ -1395,7 +1395,7 @@ 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
[
- mathcal I_calI_{J/S}
+ I_{J/S}
=
(f_*OO_{S^{up}}
(-T^{up}|_{S^{up}}))^{inv}.
@@ -1409,11 +1409,11 @@ characteristic zero invariants form a direct summand.
Apply finite-map duality with trace and then take invariants:
[
- mathcal Hom_S(calHom_S(mathcal I_calI_{J/S},omega_S)
+ Hom_S( I_calI_{J/S},omega_S)
=
(f_*omega_{S^{up}}
(T^{up}|_{S^{up}}))^{inv}
- simeqOO_S(aS).
+ ~=OO_S(aS).
]
The last isomorphism comes from Cartier adjunction upstairs
and descent of the line of S. Finite-map duality applies
@@ -1424,10 +1424,10 @@ 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_PP.
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
+map Sto Dtimes_PP. 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
@@ -1441,33 +1441,33 @@ contained in
]
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
+representable projective morphism h:StoP; 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.
+T={P}*OO_S over P.
This coherent algebra is generated as a module by vector
-bundles on mathcal P.calP. Indeed, decompose it by the finitely
+bundles on P. 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
+The resulting vector-bundle surjection embeds T in
+a vector bundle over P. Since T is
proper over the base, it remains closed in the projective
-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,
+completion. This completion is Y and gives the map
+g:StoY from the reduced divisor.
+The Stein construction yields g_*OO_S=OO_{T},
and the line identities follow from those already established.
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;
+This construction does not use a map qStoY;
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 PcalP
+The finite representable map Sto Dtimes_PP
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
@@ -1482,33 +1482,33 @@ quasi-projective scheme as required.
Our goal is to lift functions and a transverse parameter through
every infinitesimal neighborhood of the divisor constructed in
Proposition~prop:signed-construction. We recall its data.
-(mathcal P=calP=sqrt[ell]{OO_{PP^s}(1)}) is a root gerbe, (C) is
+(P=[ell]{OO_{PP^s}(1)}) is a root gerbe, (C) is
its tautological line bundle, and
[
- Sxrightarrow{g}mathcal YcalYxrightarrow{p}mathcal P,calP,
+ Sxrightarrow{g}Y{p}P,
h=pcirc g.
]
Here (g) is representable and projective, and (p) is a smooth
projective bundle. The Stein factor of (h) is finite over
-(mathcal PcalP); it has been embedded as (mathcal TcalTsubsetmathcal YcalY)
-so that (g_*OO_S=OO_{mathcal T}calT). The reduced Cartier
-divisor (Ssubsetmathcal XcalX) has dimension (N=n-1). If
+(P; it has been embedded as (T YcalY)
+so that (g_*OO_S=OO_{T} The reduced Cartier
+divisor (SsubsetX has dimension (N=n-1). If
(J=(Scap T)_{red}), then (S,J) are Du Bois,
-(mathcal I_calI_{J/S}) is maximal Cohen--Macaulay, and
+(I_{J/S}) is maximal Cohen--Macaulay, and
- mathcal B:=calB:=
- mathcal Hom_S(calHom_S(mathcal I_calI_{J/S},omega_S)
- simeq OO_S(aS)=g^*C^a,
+ B:=calB:=
+ Hom_S( I_calI_{J/S},omega_S)
+ ~= OO_S(aS)=g^*C^a,
OO_S(S)=g^*C,
for an integer (a>0). Off (T), the fixed ambient isomorphism
-(omega_{mathcal X}calX(S)simeqOO_{mathcal X}calX(aS)) supplies these
+(omega_{X}~=OO_{X} supplies these
identifications by adjunction.
-Put (I=OO_{mathcal X}calX(-S)). For a separated quasi-projective
-scheme etale chart (Utomathcal YcalY) of finite type, write
-(S_U=Stimes_{mathcal Y}calYU) and again (g:S_Uto U)
+Put (I=OO_{X}-S)). For a separated quasi-projective
+scheme etale chart (UtoY of finite type, write
+(S_U=Stimes_{Y} and again (g:S_Uto U)
for the induced projective morphism. Denote by (qS_U) the
unique extension of (S_Uto S) to the nilpotent thickening
(qS). These extensions are compatible in (q).
@@ -1528,8 +1528,8 @@ the transition
→ g_*(I^j/I^{j+k})
]
is surjective as a map of sheaves of abelian groups. Its kernel is
-(OO_{mathcal T_U}calT_U ⊗ C^{-j-k}), where
-(mathcal T_U=calT_U=mathcal TcalTtimes_{mathcal Y}calYU).
+(OO_{T_U} ⊗ C^{-j-k}), where
+(T_U= TcalTtimes_{Y}
The statements are compatible with further etale changes of
charts. On an affine chart, these transitions are also
surjective on global sections.
@@ -1581,7 +1581,7 @@ consists of the sheaf (F_0M) in degree zero.
We now identify a Hodge module whose lowest piece contains the
coherent lifting obstructions. On each chart (U), write
-(J_U=Jtimes_{mathcal Y}calYU).
+(J_U=Jtimes_{Y}
For (j:S_U∖ J_U↪ S_U), put
[
A^bullet_U=mathbf DbfD!(
@@ -1596,19 +1596,19 @@ perverse cohomology, and (g_*) in the definition of (M_U)
is the derived Hodge-module direct image. Later, (g_+) will
denote the corresponding direct image of differential modules.
These constructions commute with etale restriction. Thus the
-(M_U) form a system (M) supported on (mathcal TcalT).
+(M_U) form a system (M) supported on (T
The following identifications hold compatibly on the charts:
[
- F_{<0}A_{0,U}=0, F_0A_{0,U}=mathcal B|_calB|_{S_U},
+ F_{<0}A_{0,U}=0, F_0A_{0,U}=B|_calB|_{S_U},
F_{<0}M_U=0,
- F_0M_U=R^1g_*(mathcal B|_calB|_{S_U}).
+ F_0M_U=R^1g_*(B|_calB|_{S_U}).
]
In a smooth ambient embedding (S_U↪ V) of
codimension (c), the underlying module of (A_{0,U}) is
-(mathcal H^c_calH^c_{S_U}(omega_V)), localized off (J_U).
+(H^c_{S_U}(omega_V)), localized off (J_U).
The lowest-piece inclusion is the adjunction, or Ext-to-support,
inclusion off (J_U), extended by meromorphic localization.
@@ -1620,12 +1620,12 @@ will differentiate classes in that inclusion. We first compute
the sheaf by duality. Du Bois
comparison applied to the difference triangle for the constant
objects of (S_U) and (J_U) identifies its degree-zero
-graded de Rham complex with (mathcal I_calI_{J_U/S_U}).
+graded de Rham complex with (I_{J_U/S_U}).
Coherent duality and the maximal-Cohen--Macaulay property give
Gr^F_kDR(A^bullet_U)=0quad(k<0),
- Gr^F_0DR(A^bullet_U)=mathcal B|_calB|_{S_U}[0].
+ Gr^F_0DR(A^bullet_U)=B|_calB|_{S_U}[0].
Here ([N]) cancels the dimension shift of the dualizing
complex. In the smooth case, our convention gives
@@ -1641,13 +1641,13 @@ Spencer complex consisting only of its degree-zero term. The
spectral sequence for perverse truncation has a single nonzero
row at that index, and therefore gives
[
- mathcal H^icalH^iGr^F_qDR(A^bullet_U)
+ H^i
=F_q,{}^p!H^iA^bullet_U.
]
If (q<0), this contradicts Equation~(hg:derived-lowest).
It follows that all these objects have (F_{<0}=0). The same
argument at index zero yields
-(F_0A_{0,U}=mathcal B|_calB|_{S_U}), while the lowest pieces of
+(F_0A_{0,U}=B|_calB|_{S_U}), while the lowest pieces of
the other perverse cohomology objects vanish. This passage
does not require the shifted constant complex to be perverse.
@@ -1657,7 +1657,7 @@ identifies the dual of the reduced constant complex with
RGamma_{[S_U]}(omega_V)[c].
]
The degree-zero module is the first nonzero support cohomology,
-(mathcal H^c_calH^c_{S_U}(omega_V)). The support of (J_U) is locally
+(H^c_{S_U}(omega_V)). The support of (J_U) is locally
set-theoretically principal in (S_U) by the geometric
construction. Lift a local defining function to (V) and
invert it. This exact meromorphic localization realizes
@@ -1685,10 +1685,10 @@ direct image. To identify (F_0M_U), compute (g_+) by a
graph embedding followed by a smooth
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
+(B|_calB|_{S_U}) on the graph. Projective strictness yields
[
F_{<0}M_U=0,
- F_0M_U=R^1g_*(mathcal B|_calB|_{S_U}),
+ F_0M_U=R^1g_*(B|_calB|_{S_U}),
]
and identifies this sheaf as a coherent subsheaf of the unfiltered
degree-one direct image. All constructions in this identification
@@ -1705,17 +1705,17 @@ symbol map, and will apply the following vanishing to that kernel.
## Negative-twist vanishing
The support of the system (M) just constructed is finite over
-(mathcal PcalP). We prove the required vanishing for any system
+(P We prove the required vanishing for any system
with this support property.
Let (M) be a compatible system of mixed Hodge modules in perverse
-degree zero on the scheme etale charts of (mathcal YcalY). Suppose
-its support is finite over (mathcal PcalP). Then, for every integer
+degree zero on the scheme etale charts of (Y Suppose
+its support is finite over (P Then, for every integer
(k), every positive integer (b), and every (j<0),
[
- mathbb H^jbbH^j!(
- mathcal Y,calY,Gr^F_kDR_{mathcal Y}calY(M) ⊗ p^*C^{-b}
+ H^j!(
+ Y,Gr^F_kDR_{Y} ⊗ p^*C^{-b}
)=0.
]
Here the graded de Rham complex is the descended coherent complex.
@@ -1723,16 +1723,16 @@ Here the graded de Rham complex is the descended coherent complex.
Proof.
The line (C) lives on the root gerbe, so we first push the
-complex to (mathcal PcalP). We will then pull back to projective
+complex to (P We will then pull back to projective
space, where (C) becomes (OO(1)) and ordinary
Kodaira--Saito vanishing applies. Since (p) is finite on the
support of (M), its direct image is concentrated in perverse
-degree zero; call the resulting system on (mathcal PcalP) (M').
+degree zero; call the resulting system on (P (M').
On ordinary scheme
charts, projective strictness gives
- Rp_*Gr^F_kDR_{mathcal Y}calY(M)
- simeq Gr^F_kDR_{mathcal P}calP(M').
+ Rp_*Gr^F_kDR_{Y}
+ ~= Gr^F_kDR_{P}').
For the desired global hypercohomology vanishing, this identity must
hold for the descended complexes. We verify that compatibility
@@ -1740,21 +1740,21 @@ before passing to a finite cover.
For right differential modules, use the transfer bimodule
[
- mathcal D_calD_{mathcal YcalYtomathcal P}calP
+ D_{Y P}calP
=
- OO_{mathcal Y}calY ⊗ _{p^{-1}OO_{mathcal P}calP}
- p^{-1}mathcal D_calD_{mathcal P}calP,
+ OO_{Y} ⊗ _{p^{-1}OO_{P}}
+ p^{-1}D_{P},
]
with its order filtration. Derived tensor over
-(mathcal D_calD_{mathcal Y}calY), followed by derived direct image,
+(D_{Y}, followed by derived direct image,
forms the filtered direct image; equivalently, these operations
can be carried out on Rees modules. De Rham on the base is a
-further derived tensor with (OO_{mathcal P}calP), filtered from
+further derived tensor with (OO_{P}, filtered from
degree zero. The filtered Spencer resolution, associativity of
derived tensor, and projection formula identify this with direct
image of the de Rham complex upstairs. Indeed, tensoring the
transfer module over the base differential operators with the
-base structure sheaf gives (OO_{mathcal Y}calY).
+base structure sheaf gives (OO_{Y}
These sheaf constructions are canonical on the etale site, and the
Spencer resolutions are locally free over differential operators.
@@ -1770,14 +1770,14 @@ apply to the finite representable map below.
To pass from the gerbe to a scheme, use the representable finite
surjective morphism
[
- v:PP^s_z→mathcal PcalP
+ v:PP^s_z→P
]
defined by the power map
-([z_0:...:z_s]mapsto[z_0^ell:...:z_s^ell])
+([z_0:...:z_s]|->[z_0^ell:...:z_s^ell])
and the root (OO_{PP^s_z}(1)); thus
(v^*C=OO_{PP^s_z}(1)). Trivial roots on the standard affine
opens of (PP^s) give scheme etale charts
-(check U_itomathcal PcalP). The restriction of (v) to
+(check U_itoP The restriction of (v) to
(z_ine0) factors through (check U_i). On each such
chart take the derived Hodge-module inverse image of (M')
and then its degree-zero perverse cohomology. Thus the object
@@ -1838,7 +1838,7 @@ Equation~(hg:graded-direct-image) and projection formula
now exhibits the hypercohomology in the statement as a direct
summand of
[
- mathbb H^jbbH^j!(
+ H^j!(
PP^s_z,Gr^F_kDR(H) ⊗ OO_{PP^s_z}(-b)
).
]
@@ -1871,7 +1871,7 @@ Define
]
The first graded layer is
[
- E_{j,1}=OO_{mathcal T_U}calT_U ⊗ C^{-j}.
+ E_{j,1}=OO_{T_U} ⊗ C^{-j}.
]
For adjacent lengths, the exact sequence has kernel
(g_*OO_{S_U} ⊗ C^{-j-k}), and its connecting homomorphism
@@ -1891,11 +1891,11 @@ factors through the length-one layer; for (k=1) this
factorization is immediate. More precisely, these factorizations
give additive maps
[
- D_{k,j}:OO_{mathcal T_U}calT_U ⊗ C^{-j}
+ D_{k,j}:OO_{T_U} ⊗ C^{-j}
→ R^1g_*OO_{S_U} ⊗ C^{-j-k}.
]
Together they define (D_k) on the graded algebra
-(⊕_{jgeq0}OO_{mathcal T_U}calT_U ⊗ C^{-j}), with
+(⊕_{jgeq0}OO_{T_U} ⊗ C^{-j}), with
graded shift (k); no (OO_U)-linearity is asserted.
Its values are computed by choosing local
lifts and taking Cech differences. For a product, this
@@ -1906,17 +1906,17 @@ values in the graded module whose degree-(j) term is
(R^1g_*OO_{S_U} ⊗ C^{-j}).
Restrict its degree-zero part along
-(OO_UtoOO_{mathcal T_U}calT_U). This derivation factors
+(OO_UtoOO_{T_U} This derivation factors
through (Ω_U^1), defining a section of
(T_U ⊗ R^1g_*OO_{S_U} ⊗ C^{-k}). As the chart
varies, these sections descend to
ein H^0!(
- mathcal Y,T_calY,T_{mathcal Y}calY ⊗ R^1g_*OO_S ⊗ C^{-k}
+ Y,T_calY,T_{Y} ⊗ R^1g_*OO_S ⊗ C^{-k}
)
=
H^0!(
- mathcal Y,T_calY,T_{mathcal Y}calY ⊗ F_0M ⊗ C^{-(a+k)}
+ Y,T_calY,T_{Y} ⊗ F_0M ⊗ C^{-(a+k)}
).
The equality follows from Equation~(hg:geometric-lines),
@@ -1953,17 +1953,17 @@ Thus (D_k(t_lambda)) is represented by
target line bundles; (e_{ij,lambda}) and (eta_{ij})
are their coefficients on the reduction. Denote the induced
frame of
-(mathcal B|_calB|_{S_U}) by (sigma=y^{-a}).
+(B|_calB|_{S_U}) by (sigma=y^{-a}).
Off (J_U), the fixed adjunction isomorphism gives
[
omega_{(k+1)S_U}
- simeqOO_{mathcal X}calX((a+k)S)|_{(k+1)S_U}.
+ ~=OO_{X}+k)S)|_{(k+1)S_U}.
]
Via this isomorphism, the local equation (y_i) defines a
dualizing generator, denoted (b_i=y_i^{-(a+k)}). This
notation denotes a frame of the line bundle
-(OO_{mathcal X}calX((a+k)S)) restricted to the thickening;
+(OO_{X}+k)S)) restricted to the thickening;
it does not invert the nilpotent function (y_i) in its
structure sheaf. Realize the dualizing sheaves in support cohomology. Under
the trace inclusion
@@ -1985,12 +1985,12 @@ We next put both errors in the same differential module. Embed
(Zsubsetoverlineoverline Z=Z=PP^m). The graph of the reduced map
(g) is closed in (overlineoverline ZZtimes U), since it is proper
over (U). Let (iota:S_U↪overlineoverline ZZtimes U)
-be this reduced graph embedding, and let (mathcal N_calN_Gamma)
+be this reduced graph embedding, and let (N_
be the underlying unfiltered differential module of
(iota_+A_{0,U}). This module is distinct from the direct-image
module (M_U) on (U). For each local graph lift (h_i),
Ext-to-support cohomology sends the dualizing section (b_i)
-to a section (delta_i(b_i)) of (mathcal N_calN_Gamma).
+to a section (delta_i(b_i)) of (N_
All the lifted graphs have the same reduced support, namely
the graph of (g), so these sections lie in one support-cohomology
module rather than in different modules for the different lifts.
@@ -2001,17 +2001,17 @@ We will prove the identity
=
-(a+k)eta_{ij}sigma
+sum_{lambda=1}^d
- (e_{ij,lambda}sigma)cdot∂_{t_lambda}.
+ (e_{ij,lambda}sigma)·∂_{t_lambda}.
On the right, each reduced dualizing section (sigma) is
-included in (mathcal N_calN_Gamma) by the lowest-piece map of
+included in (N_ by the lowest-piece map of
Lemma~hg:lowest-piece. The right differential-operator
action is taken in this graph module.
The calculation also applies when (S_U) is singular. Put
(c_0=dim Z-N). Ext-to-support identifies the dualizing
sheaf of the thickening with its annihilator submodule in
-(mathcal H^calH^{c_0}_{S_U}(omega_Z)). To see this, use the
+(H^{c_0}_{S_U}(omega_Z)). To see this, use the
support-cohomology spectral sequence. Support cohomology
vanishes below (c_0), so in total degree (c_0) the Ext
group consists of homomorphisms from the structure sheaf of
@@ -2047,7 +2047,7 @@ annihilate (b_j). The linear terms are determined by
The right action on a top differential form satisfies
[
((dt_lambda)/(t_lambda-h))
- cdot∂_{t_lambda}
+ ·∂_{t_lambda}
=(dt_lambda)/((t_lambda-h)^2).
]
The doubled pole therefore contributes the positive sign in
@@ -2074,17 +2074,17 @@ the closed graph computes this boundary; all the sheaves
in question are supported there. By
Lemma~hg:lowest-piece, the reduced cocycles in
Equation~(hg:graph-identity) represent their
-(R^1g_*mathcal BcalB) classes in (F_0M). Right
+(R^1g_*B classes in (F_0M). Right
differentiation in the (U)-coordinates acts on the
pushforward complex. Consequently the Cech classes satisfy
[
-(a+k)[eta_{ij}sigma]
+sum_{lambda=1}^d
- [e_{ij,lambda}sigma]cdot∂_{t_lambda}=0
- qquadin M_U.
+ [e_{ij,lambda}sigma]·∂_{t_lambda}=0
+ in M_U.
]
Here each bracketed class belongs to
-(R^1g_*mathcal B=F_0M_UcalB=F_0M_U); after differentiation, its
+(R^1g_*B=F_0M_U; after differentiation, its
image belongs to (F_1M_U). Thus every term of the relation
lies in (F_1M), and the (eta)-term lies in (F_0M).
The equality itself was proved in the unfiltered differential
@@ -2094,7 +2094,7 @@ implies vanishing in (F_1M). Taking the order-one symbol
shows that the global section (e) from
Equation~(hg:obstruction-section) is killed by
- T_{mathcal Y}calY ⊗ F_0M
+ T_{Y} ⊗ F_0M
→ Gr^F_1M,
after twisting by (C^{-(a+k)}).
@@ -2106,7 +2106,7 @@ Because (F_{<0}M=0), the entire complex at filtration
index one is
[
Gr^F_1DR(M)=
- [T_{mathcal Y}calY ⊗ F_0M
+ [T_{Y} ⊗ F_0M
→Gr^F_1M],
]
in degrees (-1) and (0), with differential the symbol
@@ -2130,8 +2130,8 @@ unfiltered relation gives
Injectivity of the lowest-piece inclusion and (a+k>0)
then give (D_k(y)=0). The degree-zero derivation also
vanishes: its restriction to (OO_U) is the coordinate
-error (e), and (OO_UtoOO_{mathcal T_U}calT_U) is
-surjective because (mathcal T_UcalT_U) is a closed subscheme
+error (e), and (OO_UtoOO_{T_U} is
+surjective because (T_U is a closed subscheme
of (U). The coefficient
algebra and the conormal frame (y) generate the full
graded length-one algebra locally. Thus (D_k=0) in
@@ -2139,12 +2139,12 @@ every degree, completing the induction and proving
surjectivity of all the sheaf transitions.
The kernels are the stated coherent sheaves
-(OO_{mathcal T_U}calT_U ⊗ C^{-j-k}). If (U) is
+(OO_{T_U} ⊗ C^{-j-k}). If (U) is
affine, their first cohomology vanishes, so the exact
sequences of sheaves of abelian groups give surjectivity
on global sections as well. Successive choices of lifts
then give compatible formal lifts of any chosen functions
-on (mathcal T_UcalT_U) and, after trivializing (C), of the
+on (T_U and, after trivializing (C), of the
transverse conormal frame.
@@ -2187,30 +2187,30 @@ 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 YcalY
+Choose a separated affine etale chart UtoY
near a general point of a top-dimensional component of its
-Stein image mathcal T_U.calT_U. After shrinking, C is trivial,
+Stein image T_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_UcalT_U cut out only t.
+t_1,...,t_r on T_U cut out only t.
The fiber
[
F=S_{U,t}
]
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
+sheaf I_{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
+I_{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_{X}-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
@@ -2283,7 +2283,7 @@ 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,
+ N_0|_{widehatwidehat Z}Z~=OO_{widehatwidehat Z}Z,
s_0|_{widehatwidehat Z}Z=s^ell.
]
Full faithfulness algebraizes this line-bundle trivialization
@@ -2613,19 +2613,19 @@ map, which has connected general fiber after Stein factorization.
Proof.
Shrink to an open (U) where the evaluation has finitely many etale
sheets, the parameter map is smooth, and the generated involutive
-distribution (mathcal FcalF) has constant rank. Its construction is
+distribution (F has constant rank. Its construction is
algebraic: spans and brackets stabilize at the generic point after
finitely many operations, and descend from the etale sheets. The
nonempty open parts of the integral parameter fibers are connected.
A chain step moves between two points in the image of one such
open fiber. We use only steps lying in (U), so each step remains
-in one analytic leaf of (mathcal FcalF).
+in one analytic leaf of (F
For (xin U), endpoints of chains of at most (k) steps form a
constructible set, by the algebraic fiber-product construction and
Chevalley's theorem. Every endpoint lies in the analytic leaf
through (x). A locally closed smooth variety contained in an
-analytic leaf has dimension at most (rkmathcal FcalF). Indeed, in a
+analytic leaf has dimension at most (rkF Indeed, in a
Frobenius chart the leaf meets at most countably many plaques: use
finite plaque chains in a countable foliation atlas. The transverse
coordinate map on any connected smooth piece then has countable
@@ -2646,18 +2646,18 @@ that closure to equal (V). At a general
diagonal point, smoothness of the parameter map identifies its
vertical tangent with the corresponding web tangent. Every web
tangent is consequently tangent to (V), and so is every bracket. Hence
-(dim Vgeqrkmathcal FcalF), while the reverse inequality was proved
+(dim VgeqrkF, while the reverse inequality was proved
above. A plaque is thus open in (V). The equations of (V),
pulled back to the connected immersed leaf, vanish on a nonempty
open and hence on the entire leaf. Its closure is exactly (V).
-These invariant subvarieties of dimension (rkmathcal FcalF) have countably many
+These invariant subvarieties of dimension (rkF have countably many
Hilbert or Chow parameter spaces. Tangency on the regular locus is
an algebraic condition after stratification, so one such family
dominates. An invariant irreducible variety meeting (U) contains
the leaf through any of its points in (U): the tangent vector
fields preserve its reduced ideal, first on its smooth locus and
-then everywhere by closure. A leaf closure of dimension (rkmathcal FcalF) is
+then everywhere by closure. A leaf closure of dimension (rkF is
therefore unique through a general point. Assigning that closure
gives the desired rational map. Resolve its graph and take the
Stein factorization.
@@ -2674,8 +2674,8 @@ 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),
- iota_{[v,w]}T=mathcal L_v(calL_v(iota_wT)-iota_w(mathcal L_vT).calL_vT).
+ L_v T=d(iota_vT)+iota_v(dT),
+ iota_{[v,w]}T=L_v(-iota_w(L_vT).
]
Pullbacks by dominant morphisms and restrictions to almost every
smooth parameter fiber are defined by the same local potentials.
@@ -2827,7 +2827,7 @@ write (A_V(E)=a(E;V,0)) for log discrepancy.
For a fixed (T) and a fixed (M), the set
[
- {nu_E(T): A_V(E)leq M}
+ {nu_E(T): A_V(E)<= M}
]
satisfies the ascending chain condition.
@@ -2839,13 +2839,13 @@ increasing sequence exists, and discard initial terms so its
members exceed a fixed positive number (c).
Skoda integrability and change of variables give
- nu_E(T)leq A_V(E)nu_x(T)
+ nu_E(T)<= A_V(E)nu_x(T)
at a general point (x) of the center. Indeed every exponent
smaller than (1/nu_x(T)) is locally integrable, whereas
integrability after pullback imposes the corresponding
discrepancy bound. Hence all centers lie in the fixed proper Siu
-locus ({nu_x(T)geq c/M}), which is algebraic by Siu
+locus ({nu_x(T)>= c/M}), which is algebraic by Siu
analyticity and projectivity [cite]. Passing to a
subsequence, all centers lie in one irreducible component of
this locus.
@@ -2863,7 +2863,7 @@ Siu locus. Write (T=c_D[D]+T') with (T') positive and with
zero generic Lelong number along (D). The integers
(ord_E(D)) are bounded: the fixed effective divisor (D) has
positive log canonical threshold, and
-(lct(D)ord_E(D)leq A_V(E)).
+(lct(D)ord_E(D)<= A_V(E)).
Pass to a constant order. The residual numbers (nu_E(T'))
are still strictly increasing, and after discarding a term have
a positive lower bound. Equation~(ex:skoda-valuation)
@@ -2878,13 +2878,13 @@ cite[Theorems~5.11, 6.27, and 14.2]{Demailly2012}. On a smooth projective variet
is a real algebraic class, one can choose integral line bundles
(L_k) such that (c_1(L_k)-k{T}) stays in a bounded,
uniformly sufficiently ample set and
-(L_k ⊗ mathcal J(kT)calJ(kT)) is globally generated. Round the
+(L_k ⊗ J(kT) is globally generated. Round the
coefficients of (k{T}) in a fixed integral basis and add a
fixed sufficiently positive integral class. Nadel vanishing and
Castelnuovo--Mumford regularity give generation. For every
divisorial valuation,
- ord_Emathcal J(kT)calJ(kT)leq knu_E(T).
+ ord_EJ(kT)<= knu_E(T).
The local Bergman weight is bounded below by the original weight
up to a constant; this inequality persists after pullback and
@@ -3026,12 +3026,12 @@ pseudo-effective cone at (K_Y), and put
on higher smooth base models are movable classes by
pseudo-effective/movable duality cite[Theorem~0.2]{BDPP2013}.
They annihilate base-exceptional divisors. Since (S) is positive,
-(etacdot{S}geq0). Applying (alpha) to
+(eta·{S}geq0). Applying (alpha) to
(ex:descent-domination) gives the reverse inequality,
-because (alpha(K_Y)=0). Hence (etacdot{S}=0).
+because (alpha(K_Y)=0). Hence (eta·{S}=0).
If (c_D[D]) is a positive Siu component of (S), both
(S-c_D[D]) and ([D]) are positive. It follows that
-(etacdot D=0), or equivalently (alpha(T_BD)=0).
+(eta· D=0), or equivalently (alpha(T_BD)=0).
Before applying multiplier approximation, we need to control the
divisorial part of this descended current. Only finitely many prime
@@ -3058,7 +3058,7 @@ contradicts Corollary~ex:supporting.
We claim that
- K_Bcdoteta<0.
+ K_B·eta<0.
The smooth general fiber (F) of (h) is of general type by
Proposition~prop:general-type. The relative-positivity
@@ -3072,7 +3072,7 @@ smooth projective, the boundary is zero, and we take (M=0).
Invariance of plurigenera on the smooth locus identifies the
generic-fiber Kodaira dimension with that of (F). Thus
- kappa(W,K_W-h^*K_B)geq kappa(F,K_F)=n-b>0.
+ kappa(W,K_W-h^*K_B)>= kappa(F,K_F)=n-b>0.
This is an established general-type-fiber theorem, not another
use or strengthening of Theorem~asm:iitaka.
@@ -3080,58 +3080,58 @@ Choose compatible canonical divisors and set
(P=K_Y-T_B(K_B)=p_*(K_W-h^*K_B)).
For every sufficiently divisible (m), pushing forward an
effective divisor
-(div_W(varphi)+m(K_W-h^*K_B))
+(div_W(phi)+m(K_W-h^*K_B))
gives
-(div_Y(varphi)+mPgeq0).
+(div_Y(phi)+mPgeq0).
The resulting injection of sections preserves their ratios,
so (kappa(Y,P)geq1). Corollary~ex:supporting now gives
-(0<alpha(P)=-K_Bcdoteta), proving
+(0<alpha(P)=-K_B·eta), proving
(ex:negative-base).
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).
+principalize (J(kS) by (r_k:B_kto B).
If (F_k) is its divisor, the class
[
P_k=(r_k^*L_k-F_k)/(k)
]
is nef. Put (eta_k=T_{B_k}^*alpha). The class (eta_k)
is movable and (F_k) is effective, so
-(0leq P_kcdoteta_kleq (L_kcdoteta)/k).
-Since (etacdot{S}=0) and (L_k-k{S}) stays in a
+(0leq P_k·eta_kleq (L_k·eta)/k).
+Since (eta·{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}),
- K_{B_k}cdoteta_k=K_Bcdoteta<0.
+ 0leq P_k·eta_k=O(k^{-1}),
+ K_{B_k}·eta_k=K_B·eta<0.
]
Fix an ample divisor (H) on (B), and write
-(delta=-K_Bcdoteta>0). On (B_k), choose an ample
+(delta=-K_B·eta>0). On (B_k), choose an ample
divisor (H_k) and a sufficiently small rational
(ε_k>0) so that the ample rational class
[
Q_k=P_k+k^{-1}r_k^*H+ε_k H_k
]
-satisfies (Q_kcdoteta_k=O(k^{-1})). This is possible
-because (eta_kcdot r_k^*H=etacdot H) is fixed.
+satisfies (Q_k·eta_k=O(k^{-1})). This is possible
+because (eta_k· r_k^*H=eta· H) is fixed.
Approximate (eta_k) by a positive sum
(gamma_k=sum_{i=1}^{N_k} a_i[C_i]), with (a_i>0),
of covering-curve classes,
closely enough that
[
- Q_kcdotgamma_kleq C/k,
- -K_{B_k}cdotgamma_kgeqdelta/2
+ Q_k·gamma_kleq C/k,
+ -K_{B_k}·gamma_kgeqdelta/2
]
for a constant (C) independent of (k). Keep only terms
-with (K_{B_k}cdot C_i<0). Their total anticanonical
+with (K_{B_k}· C_i<0). Their total anticanonical
degree is at least (delta/2), while their total
(Q_k)-degree is at most (C/k). Taking the weighted
average of the ratios therefore gives one covering curve
(Gamma_k) with
[
- frac{Q_kcdotGamma_k}{-K_{B_k}cdotGamma_k}
- leq(2C)/(delta k).
+ frac{Q_k·Gamma_k}{-K_{B_k}·Gamma_k}
+ <=(2C)/(delta k).
]
The quantitative bend-and-break estimate
cite[Theorem~5]{MiyaokaMori1986} gives a rational curve
@@ -3141,24 +3141,24 @@ 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}),
- Hcdot r_{k*}R_k=O(1).
+ P_k· R_k=O(k^{-1}),
+ H· r_{k*}R_k=O(1).
Shrink each parameter space so that the curve class and its
contacts with the exceptional divisors and the strict transforms
of the Siu divisors of (S) are constant. The general parametrized
member is free in characteristic zero. Consequently
-(K_{B_k}cdot R_kleq-2). Put (barbar R_k=r_R_k=r_{k*}R_k).
-Its bounded (H)-degree bounds (K_Bcdotbarbar R_kR_k), and
+(K_{B_k}· R_kleq-2). Put (barbar R_k=r_R_k=r_{k*}R_k).
+Its bounded (H)-degree bounds (K_B·barbar R_kR_k), and
effectivity of the relative canonical divisor gives
- 0leq K_{B_k/B}cdot R_k
- =K_{B_k}cdot R_k-K_Bcdot r_{k*}R_k=O(1).
+ 0leq K_{B_k/B}· R_k
+ =K_{B_k}· R_k-K_B· r_{k*}R_k=O(1).
In particular, the upper bound is independent of (k).
If an exceptional prime (E) has coefficient
(a_E=A_B(E)-1) in (K_{B_k/B}), a positive contact with
-it contributes (a_E(Ecdot R_k)) to this bounded sum.
+it contributes (a_E(E· R_k)) to this bounded sum.
Both factors are positive integers. Thus the number of
exceptional primes met by (R_k), their log discrepancies
(A_B(E)=a_E+1), and their contact degrees are uniformly
@@ -3174,23 +3174,23 @@ nonnegative. Using (ex:multiplier-order) and
(ex:small-curves) gives
- 0&leq {S}cdotbarbar R_kR_k
- -sum_Enu_E(S),Ecdot R_k
- &leq {S}cdotbarbar R_k-(F_k/k)R_k-(F_k/k)cdot R_k
- leq O(k^{-1}).
+ 0&<= {S}·barbar R_kR_k
+ -sum_Enu_E(S),E· R_k
+ &<= {S}·barbar R_k-(F_k/k)R_k-(F_k/k)· R_k
+ <= O(k^{-1}).
The sum includes exceptional primes and those strict
divisorial components; terms with zero contact are irrelevant.
For the last bound, the expression involving (F_k) equals
-(P_kcdot R_k+({S}-L_k/k)cdotbarbar R_kR_k).
+(P_k· R_k+({S}-L_k/k)·barbar R_kR_k).
The first term is (O(k^{-1})) by (ex:small-curves);
the second has the same bound because (L_k-k{S}) stays
bounded and the (H)-degrees of (barbar R_kR_k) are bounded.
Bounded ample degree gives only finitely many numerical
classes of the integral cycles (barbar R_kR_k). Pass to one
-class, so ({S}cdotbarbar R_kR_k) is now one fixed real
+class, so ({S}·barbar R_kR_k) is now one fixed real
number. The current (S) on (B) also remains fixed. By
Lemma~ex:valuation-acc, the sums in
(ex:zero-gap) belong to an ACC set: there are a bounded
@@ -3264,12 +3264,12 @@ fixed resolution carrying (T). Choose on the fixed
resolution a sufficiently positive line bundle (H).
For sufficiently divisible (k), multiplier approximation
gives a globally generated
-(OO(kp^*K_Y+H) ⊗ mathcal J(kT)calJ(kT)).
+(OO(kp^*K_Y+H) ⊗ J(kT)
Pull this system to (V). For the finite collection of primes
on the common resolution whose coefficients we are comparing,
choose a general member (D_k) of the underlying effective
divisor system. At every such prime (E), it has multiplicity
-(ord_Emathcal J(kT)calJ(kT)); global generation after removal
+(ord_EJ(kT); global generation after removal
of the ideal ensures no additional generic vanishing.
Here the notation (H) also denotes its pullback to (V).
@@ -3290,7 +3290,7 @@ applied with both signs makes it zero. Since
(p_t^*D'_k) is effective,
[
coeff_E D_k
- geq kcoeff_E J_t
+ >= kcoeff_E J_t
+coeff_E(H-p_t^*H_t).
]
The last coefficient is fixed for this model and (t).
@@ -3307,7 +3307,7 @@ families of curves on (Y_{t_j}), with smooth projective
domains (C_j), for which both
[
g(C_j),
- frac{(K_{t_j}+t_jA_{t_j})cdot C_j}{mu_{t_j}}
+ frac{(K_{t_j}+t_jA_{t_j})· C_j}{mu_{t_j}}
]
are bounded by a fixed constant. Degrees mean degrees on
the domains, or equivalently intersection with their
@@ -3355,24 +3355,24 @@ 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,
- q_t^*K_Ycdot C_tgeq0.
+ K_{U_t}· C_t=2g(C_t)-2,
+ q_t^*K_Y· C_tgeq0.
]
The latter inequality follows from pseudoeffectivity and
the covering property. Thus
[
- 0leq (K_{U_t}-q_t^*K_Y)cdot C_tleq 2g_0-2.
+ 0leq (K_{U_t}-q_t^*K_Y)· C_tleq 2g_0-2.
]
Each exceptional contact contributes
-((r_Za(E;Y,0)-1)(Zcdot C_t)) to this bounded sum.
+((r_Za(E;Y,0)-1)(Z· C_t)) to this bounded sum.
The first factor is at least (1/m), and the second is
a positive integer. Formula~(ex:ramification-cost)
therefore bounds the number of contacted exceptional primes,
their log discrepancies, their ramification indices, and
their contact degrees. In particular the integers
-(r_Z(Zcdot C_t)) are uniformly bounded. Also
+(r_Z(Z· C_t)) are uniformly bounded. Also
[
- q_t^*K_Ycdot C_tin m^{-1}Zcap[0,2g_0-2],
+ q_t^*K_Y· C_tin m^{-1}Zcap[0,2g_0-2],
]
so these degrees take only finitely many values.
@@ -3394,11 +3394,11 @@ fiber. As in (ex:zero-gap), its degree is the first
difference below and is nonnegative. Consequently
- 0&leq q_t^*K_Ycdot C_t
+ 0&<= q_t^*K_Y· C_t
-sum_{Z {rm exceptional}}
- r_Znu_E(T),Zcdot C_t
- &leq K_tcdot C_t
- leq (K_t+tA_t)cdot C_t
+ r_Znu_E(T),Z· C_t
+ &<= K_t· C_t
+ <= (K_t+tA_t)· C_t
=O(mu_t).
@@ -3408,7 +3408,7 @@ curves cover the total space.
On the fixed smooth resolution (p:Wto Y), one has
[
a(E;W,0)=a(E;Y,0)
- -ord_E(K_W-p^*K_Y)leq a(E;Y,0),
+ -ord_E(K_W-p^*K_Y)<= a(E;Y,0),
]
because the relative canonical divisor is effective.
Thus Lemma~ex:valuation-acc applies to the one fixed
@@ -3416,8 +3416,8 @@ current (T) on (W), with a fixed discrepancy bound.
The sums in (ex:bounded-gap) form an ACC set: their
Lelong numbers belong to this ACC set, and both the number
of terms and their integral multiplicities
-(r_Z(Zcdot C_t)) are bounded. Pass to a fixed value of
-(q_t^*K_Ycdot C_t). Since (mu_tto0), the
+(r_Z(Z· C_t)) are bounded. Pass to a fixed value of
+(q_t^*K_Y· C_t). Since (mu_tto0), the
exact-zero argument of (ex:zero-gap) gives a family
for which the left gap is zero.
@@ -3467,7 +3467,7 @@ cite[Theorem~1.9(ii), equivalently Theorem~4.1(ii)]
{Birkar2012Special}. Discard the finitely many divisorial
contractions and write the flips as (V_idashrightarrow V_{i+1}/Z_i),
with scaling numbers (lambda_i>0) tending to zero.
-Fix a component (S_1) of (lfloorDelta_1rfloor), write
+Fix a component (S_1) of (floor(Delta_1)), write
(S_i) for its normal strict transform, and let (T_i) be the
normalization of its image in (Z_i). Since the ambient contraction
is small, (S_ito T_i) is projective birational. The standard
@@ -3520,10 +3520,10 @@ 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
[
- (D_i+lambda_i C_i)cdot R=0,
- D_icdot R<0.
+ (D_i+lambda_i C_i)· R=0,
+ D_i· R<0.
]
-It follows that C_icdot R>0. Every smaller coefficient of
+It follows that C_i· 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.
@@ -3539,8 +3539,8 @@ 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.
+ (D_i+lambda_j C_i)· R
+ =(lambda_j-lambda_i)C_i· Rle0.
]
Thus all preceding steps are nonpositive for the adjoint with
this fixed coefficient lambda_j. On a common resolution of
@@ -3668,7 +3668,7 @@ Suppose such a family exists, and denote its birational maps by
Choose one map (phi_{a_0}) in this family and use it to identify
(T_2) birationally with (T_1). The maps
(g_a=phi_{a_0}^{-1}circphi_a) are then birational selfmaps of
-(T_1). The evaluation ((a,u)mapsto(u,g_a(u))) is dominant onto
+(T_1). The evaluation ((a,u)|->(u,g_a(u))) is dominant onto
(T_1times T_1); in particular, the images of a general point under
these selfmaps are dense.
@@ -3750,7 +3750,7 @@ 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}.
+is primary for the maximal ideal of O_{Z,eta_V}.
All powers of ideals below are ordinary powers.
## Numerical subadjunction on the center
@@ -3772,8 +3772,8 @@ 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}.
+ K_{V^*}·(rho^*P)^{f-1}
+ <= (K_Z+Θ)|_V· 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
@@ -3836,7 +3836,7 @@ 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
+O_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
@@ -3873,12 +3873,12 @@ 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.
+ M_{V'}·(P')^{f-1}
+ =M_{widehatwidehat V'}V'·(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.
+B_{V'}·(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
@@ -3898,8 +3898,8 @@ Q-Cartier. Intersect Equation~(common:subadjunction-cbf) with
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}.
+ (degeta)D_V·(nu^*P)^{f-1}
+ >=(degeta)K_{V^nu}·(nu^*P)^{f-1}.
]
Finally the pushforward of K_{V^*} to V^nu is its compatible
canonical cycle. All additional divisors on the resolution have
@@ -3921,20 +3921,20 @@ degree of the center and the coefficient of that boundary.
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
+0neH H^0(Z,O_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 m=frakm=mathfrak m_frakm_{O_{Z,eta_V}}. If
[
- mathfrak bfrakbcO_{Z,eta_V}subsetmathfrak m^hfrakm^h
+ mathfrak bfrakbO_{Z,eta_V}subsetmathfrak m^hfrakm^h
(hinZ_{>0}),
]
then
- P^fcdot Vleq (k/h)^aP^d.
+ P^f· 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
@@ -3942,24 +3942,24 @@ effective rational divisor Θsim_{Q}cP such that
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}.
+ K_{V^*}· P^{f-1}
+ <= (K_Z+cP)|_V· P^{f-1}.
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.
+The local ring R=O_{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
+Choose a general sections of H. 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.
+ =e(mathfrak q,R)frakq,R)>= e(mathfrak m^h,R)=h^a.frakm^h,R)=h^a.
]
This is the generic intersection multiplicity along V.
@@ -3989,7 +3989,7 @@ 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
moving part. Choose sufficiently many general members
-D_1,...,D_N of mathcal HcalH and put
+D_1,...,D_N of H and put
[
Θ=(lambda)/(N)(D_1+...+D_N)
sim_{Q}cP, c=klambda.
@@ -4024,12 +4024,12 @@ 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.
+ tau_{i+1}-tau_i=delta>0.
]
-For a smooth point x, let mathcal H_i(x)calH_i(x) be the entire
+For a smooth point x, let H_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)
+subspaces are nonzero and the base locus of H_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
@@ -4045,23 +4045,23 @@ point, and its entire higher-series base ideal is contained
there in
- mathcal I_calI_{V_t}^{h}, h=lceil kdelta/2rceil,
+ I_{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
+The incidence base ideal lies in I_ 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,
+ P^f· V_t&le(2/delta)^{d-f}P^d,
K_{V_t^*}P^{f-1}&le(K_Z+cP)|_{V_t}P^{f-1},
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
+Positivity of P^f· 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.
@@ -4072,7 +4072,7 @@ bundle is a morphism from the constant bundle
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
+H_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
@@ -4094,12 +4094,12 @@ 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 KcalK to the pullback of cO_Z(kP).
+K. Evaluation on Ttimes Z gives a map from the
+pullback of K to the pullback of O_Z(kP).
Let mathfrak b_Tfrakb_T be its base ideal. After trivializing the
-target line bundle, a local frame of mathcal KcalK generates
+target line bundle, a local frame of K generates
mathfrak b_T.frakb_T. We show that every frame section lies in
-mathcal I_calI_Gamma^h.
+I_
Each frame element has the form
[
s(t,z)=sum_alpha c_alpha(t)s_alpha(z),
@@ -4145,7 +4145,7 @@ 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.
+sinI_ 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
@@ -4158,7 +4158,7 @@ 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.
+I_{{t}times Z}=I_{V_t}^h locally.
The specialized frame is a basis of the entire higher kernel,
so the containment concerns its whole base ideal.
@@ -4186,7 +4186,7 @@ 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.
+contained in I_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
@@ -4210,8 +4210,8 @@ 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,
- mathcal I_calI_{V_t}^{h}mathcal O_calO_{Z_b}=mathcal I_F^calI_F^{h}
+ I_{V_t}O_{Z_b}=I_F,
+ I_{V_t}^{h}O_{Z_b}=I_F^{h}
]
near that point. Restricting the actual sections extends their
base ideal, which proves both asserted local properties.
@@ -4237,21 +4237,21 @@ 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
+V 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:V 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_NcalR_N in mathcal V.calV.
-Retain the notation mathcal V,T,gcalV,T,g after this base change.
+R_1,...,calR_1,...,R_N in V.
+Retain the notation 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.
+g_t((R_j)_t)bar is a divisor in Y for general t.
Assume
[
- g(mathcal R_j)barcalR_j)barne Yqquad(jin I).
+ g(R_j)bar Y (jin I).
]
Then, after shrinking again, one smooth nonempty open
Y^circsubset Y makes every general finite normal Stein cover
@@ -4268,7 +4268,7 @@ 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_jcalR_j in the statement.
+closures give the divisors R_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
@@ -4281,7 +4281,7 @@ parameters disappear upon shrinking.
Put
[
- B=Y_{rm sing}cup∪_{jin I}g(mathcal R_j)bar.calR_j)bar.
+ B=Y_{rm sing}cup∪_{jin I}g(R_j)bar.
]
Each closed set in this finite union is proper by hypothesis,
so Bne Y. The finite normal covers are unramified in
@@ -4405,7 +4405,7 @@ of order
]
and, when (Z) is (Q)-factorial and klt, the estimates
- P^fcdot Vle (2/delta)^{d-f}P^d,
+ P^f· Vle (2/delta)^{d-f}P^d,
K_{V^*}P^{f-1}le(K_Z+cP)|_V P^{f-1},
0<cle 2(d-f)/deltale2d/delta.
@@ -4415,7 +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^f· 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
@@ -4698,7 +4698,7 @@ point (xin Z) is
[
ε(P;x)=
inf_{substack{Csubset Z integral curvexin C}}
- (Pcdot C)/(mult_x C).
+ (P· C)/(mult_x C).
]
[Large two-slot Seshadri constant]
@@ -4777,7 +4777,7 @@ in the first slot there. Let (F) be the reduced component of the
scheme fiber through that point, and suppose (0<dim F<n+1).
Lemma~common:fiber-restriction now shows that, near that point, the
restricted higher-series base ideal has support exactly (F) and lies in the
-ordinary power (mathcal I_F^hcalI_F^h), with (h=lceil kdelta/2rceil).
+ordinary power (I_F^h, with (h=lceil kdelta/2rceil).
Because (F) is proper in the slice, the restricted series is nonzero
and its generic base ideal is primary.
@@ -4797,7 +4797,7 @@ the preceding restriction gives ordinary multiplicity at least
(hge kdelta/2). Writing (f=dim F), the lemma therefore gives
[
- P_{sl}^fcdot F
+ P_{sl}^f· F
&le(2/delta)^{n+1-f}P_{sl}^{n+1},
K_{F^*}P_{sl}^{f-1}
&le(K_{Z_{sl}}+cP_{sl})|_F
@@ -5165,11 +5165,11 @@ 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,
- xi=c_1(mathcal O_Z(1)),calO_Z(1)), M=L_1+L_2+qxi,
+ Z=mathbf P(bfP(O_W(S)_1 ⊕
+ O_W(S)_2)→calO_W(S)_2)→ Wtimes W,
+ xi=c_1(O_Z(1)), M=L_1+L_2+qxi,
]
-use the convention that the direct image of mathcal O_Z(acalO_Z(axi)
+use the convention that the direct image of O_Z(a
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,
@@ -5182,10 +5182,10 @@ curve class
]
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
+integral Cartier divisors, such that J·gamma>0 and
- pi_x^*(L+K_W/2+lambda S)cdotgamma
- le40ε Jcdotgammaqquad(0lelambdale q).
+ pi_x^*(L+K_W/2+lambda S)·gamma
+ le40ε J·gamma (0lelambdale q).
It suffices to check the two endpoints in this affine inequality.
@@ -5205,7 +5205,7 @@ characteristic tends to infinity.
We prove the theorem in the next five steps. Put
Λ=prod_{i=1}^{n-1}l_i. The flag satisfies
- [C]=Λ H^{n-1}, h_icdot C=l_iΛ H^n.
+ [C]=Λ H^{n-1}, h_i· 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
@@ -5247,7 +5247,7 @@ specialization of a characteristic-zero effective cone is used.
For a sufficiently large residual characteristic p, set
[
- N_p=lfloor p/k_0rfloor, B_p=k_0N_pL+T_0.
+ N_p=floor( p/k_0), 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
@@ -5266,8 +5266,8 @@ place in the local ring and never divides by factorials.
Put d_p=(p-k_0N_p)q, so 0le d_ple(k_0-1)q.
Among the sections fixed before reduction, choose
[
- u_0in H^0(W,cO_W(T_0)),
- v_{d_p}in H^0(W,cO_W(T_0+d_pS))
+ u_0in H^0(W,O_W(T_0)),
+ v_{d_p}in H^0(W,O_W(T_0+d_pS))
]
nonzero at the first and second base coordinates of z_*,
respectively. Their tensor product belongs to the summand of
@@ -5294,15 +5294,15 @@ 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), s=rkmathcal E=p^n,calE=p^n, R=s^2,
- U_a=H^0(W,cO_W(B_p+paS)).
+ E=F_*calE=F_*O_W(B_p), s=rkE=p^n, R=s^2,
+ U_a=H^0(W,O_W(B_p+paS)).
]
Projection formula gives an evaluation map on W'times W':
⊕_{substack{a+b=qa,bge0}}
U_a ⊗ U_b ⊗
- cO(-aS'_1-bS'_2)
- → mathcal EcalEboxtimesmathcal E.calE.
+ O(-aS'_1-bS'_2)
+ → E E.calE.
@@ -5352,32 +5352,32 @@ The Frobenius fiber product has the cartesian square
Its reduced subscheme is the diagonal W. Put
[
- mathcal L_p=calL_p=
- cO_{Delta_F}(B_{p,1}+B_{p,2}+pqS_1).
+ L_p=
+ O_{Delta_F}(B_{p,1}+B_{p,2}+pqS_1).
]
Write h=Fcircpr_1=Fcircpr_2:
Delta_Fto W'. On this fiber product,
[
- cO(pS_1)simeq h^*cO_{W'}(S')simeqcO(pS_2).
+ O(pS_1)~= h^*O_{W'}(S')~=O(pS_2).
]
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.
+ O_{Delta_F}(B_{p,1}+B_{p,2}+paS_1+pbS_2)
+ ~=L_p.
]
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.
+ (E ⊗ E)(qS')calE)(qS')~= h_*L_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 L_p. Their values at a diagonal point
+therefore lie in the image of H^0(Delta_F,L_p) in the
+corresponding fiber of h_*L_p. This proves that the rank
at every diagonal point is at most
- h^0(Delta_F,mathcal L_p).calL_p).
+ h^0(Delta_F,L_p).
For a rank-n bundle V in characteristic p, denote by A_j(V)
@@ -5390,10 +5390,10 @@ quotient of V^{ ⊗ j}. Set
u=n(p-1), e_j=rk A_j(V),
sum_{j=0}^u e_j=p^n.
]
-Filtering mathcal L_pcalL_p by powers of the ideal of the reduced
+Filtering L_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),
+ G_j=+pqS) ⊗ A_j(Ω_W^1),
0le jle u.
Indeed, in smooth local coordinates the algebra is generated by
@@ -5402,15 +5402,15 @@ 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 EcalE with pieces
+only the bundle from the second factor, filters F^*E with pieces
cite[Theorem~3.7]{Sun2008}
- cO_W(B_p) ⊗ A_j(Ω_W^1).
+ O_W(B_p) ⊗ A_j(Ω_W^1).
Complementary multiplication is a perfect pairing, giving
- A_{u-j}(V)simeq A_j(V)^vee ⊗ (det V)^{p-1}.
+ A_{u-j}(V)~= 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
@@ -5431,7 +5431,7 @@ 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.
+ mu_{max}(A_j(V))le(p-1)K_W· C+nc.
]
@@ -5478,8 +5478,8 @@ pairing Equation~(common:socle-pairing) yields
mu_{max}(A_j(V))
&= (p-1)degdet V-mu_{min}(A_{u-j}(V))
- &le(p-1)K_Wcdot C+((u-j)c)/(p-1)
- &le(p-1)K_Wcdot C+nc.
+ &le(p-1)K_W· C+((u-j)c)/(p-1)
+ &le(p-1)K_W· C+nc.
@@ -5487,7 +5487,7 @@ 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,
- mu_{max}(mathcal G_j|_C)calG_j|_C)
+ mu_{max}(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,
@@ -5497,7 +5497,7 @@ room in this common bound.
For all sufficiently large p,
[
- h^0(Delta_F,mathcal L_p)calL_p)
+ h^0(Delta_F,L_p)
le R(7^nH^n+O(1/p))<R/4.
]
The error constant is independent of the filtration index j.
@@ -5513,24 +5513,24 @@ Write the fixed flag as
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}}
+ F_k=
+ G_j|_calG_j|_{W_{n-k+1}}
(-sum_{i<k}b_i h_i).
]
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)
+ 0→F_k(-(b_k+1)h_k)calF_k(-(b_k+1)h_k)
+ →F_k(-b_kh_k)calF_k(-b_kh_k)
→
- mathcal F_k(-b_kh_k)|_calF_k(-b_kh_k)|_{W_{n-k}}
+ 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
member of the flag bounds the section space by
- h^0(W,mathcal G_j)calG_j)le
+ h^0(W,G_j)
sum_{b_1,...,b_{n-1}ge0}
- h^0(C,mathcal G_j|_C(-calG_j|_C(-sum_i b_i h_i|_C)).
+ h^0(C,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
@@ -5538,13 +5538,13 @@ 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
+sum_i b_i(h_i· 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;
+Indeed, evaluation at floor( M_p)+1 distinct points is injective;
the kernel has negative maximum slope after subtracting those
points. By Equation~(common:flag-degrees) it follows that
- h^0(W,mathcal G_j)calG_j)
+ h^0(W,G_j)
&le e_j(M_p+1)
prod_{i=1}^{n-1}
(1+(M_p)/(l_iΛ H^n))notag
@@ -5572,17 +5572,17 @@ 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)
+ 0nesigmain H^0(W'times W',Q_1 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
+(detE)^s, and the
chosen source determinant is
-cO(-m_1S')boxtimescO(-m_2S'). Thus
+O(-m_1S')boxtimesO(-m_2S'). Thus
[
- mathcal Q_i=(calQ_i=(detmathcal E)^calE)^{ ⊗ s}
- ⊗ cO_{W'}(m_iS'),
+ Q_i=( E)^calE)^{ ⊗ s}
+ ⊗ O_{W'}(m_iS'),
lambda_i=m_i/R.
]
Every column weight lies between 0 and q, so
@@ -5590,8 +5590,8 @@ Every column weight lies between 0 and q, so
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{c_1(Q_i)}
+ =frac{c_1(E)}+lambda_iS
=frac{B_p}p+(p-1)/(2p)K_W+lambda_iS,
0lelambda_ile q.
@@ -5603,12 +5603,12 @@ 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
+Thus c_1(F^*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
[
- (L+K_W/2+lambda S)cdotgammale40ε Jcdotgamma,
+ (L+K_W/2+lambda S)·gammale40ε J·gamma,
]
where pullback by pi_x is understood. This concerns only
intersections of fixed divisors and the fixed complete-intersection
@@ -5624,9 +5624,9 @@ 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
+H^0(W',Q_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
+Since J·gamma>0, it follows that
frac{ord_{x'}(tau_i)}Rle40ε+O(1/p).
@@ -5635,7 +5635,7 @@ 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
+H^0(W',Q_1) ⊗ H^0(W',Q_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
@@ -5735,7 +5735,7 @@ moving parts gives
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).
+ K_WH_0^{n-1}=KH_0^{n-1}<=(H_0^n)/(r).
To justify these intersections, recall that the maps
(Ydashrightarrow Y_t) are small by the choice of the late model.
@@ -5758,7 +5758,7 @@ small enough that continuity gives
H^nleq(2ε)^n,
- K_WH^{n-1}leq(2H^n)/(r),
+ K_WH^{n-1}<=(2H^n)/(r),
(2L+qmK)H^{n-1}leq4H^n.
@@ -5813,8 +5813,8 @@ Movable-curve duality cite[Theorem~2.2]{BDPP2013} therefore supplies an
actual covering curve class (gamma) on (widehatwidehat WW) such that
- (pi_x^*D^+)cdotgamma
- <40ε,Jcdotgamma, Jcdotgamma>0.
+ (pi_x^*D^+)·gamma
+ <40ε,J·gamma, J·gamma>0.
The strictness lets us fix an algebraic family that will survive
reduction. More precisely, the negative pairing can first be detected
@@ -5823,7 +5823,7 @@ intersection of very ample divisors on one fixed smooth birational
model of (widehatwidehat WW). Fix this model, its divisors, and the resulting
covering family. Thus the choice consists of finite algebraic data,
rather than a limiting real movable class. Positivity of
-(Jcdotgamma) follows from the strict inequality and
+(J·gamma) follows from the strict inequality and
pseudo-effectivity of (pi_x^*D^+).
To turn this curve into the witness required by
@@ -5840,10 +5840,10 @@ 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,
- Jcdotgamma>0.
+ pi_x^*Q_lambda·gamma
+ le pi_x^*D^+·gamma
+ <40ε J·gamma,
+ J·gamma>0.
]
These are the required movable inequalities. Only the two endpoint
inequalities need to be retained as finite data, since the pairing is
@@ -5896,8 +5896,8 @@ filler sections before choosing any residual characteristic.
The common theorem now rules out these fixed data. Its contradiction
compares two orders of the same nonzero determinant. After reduction,
-the Frobenius direct image (mathcal EcalE) used there has rank (p^n),
-so (mathcal EcalEboxtimesmathcal EcalE) has rank (R=p^{2n}). The fixed
+the Frobenius direct image (E used there has rank (p^n),
+so (E EcalE) has rank (R=p^{2n}). The fixed
jets give full evaluation rank (R), while the small polarization
bounds the rank on the diagonal by (R/4). Thus a nonzero maximal
minor vanishes to order at least (3R/4) at ((x',x')), where (x')
@@ -6073,11 +6073,11 @@ 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)^*),
- Lsimeqphi^*OO(1).
+ L~=phi^*OO(1).
]
The image has dimension at most (kappa(Z,A)=0), so it is a point.
A linear form nonzero at that point pulls back to a nowhere-vanishing
-section of (L). Thus (LsimeqOO_Z), an actual line-bundle
+section of (L). Thus (L~=OO_Z), an actual line-bundle
isomorphism, and (Asim_{Q}0).
@@ -6137,7 +6137,7 @@ cite[Theorem~1.5]{FujinoGongyo2014} now gives semiampleness on (X).
[Supported extension for dlt pairs over (C)]
Let ((X,Delta)) be a projective dlt pair over (C), where (Delta)
-is an effective rational divisor, and put (S=lfloorDeltarfloor).
+is an effective rational divisor, and put (S=floor(Delta)).
Assume that (K_X+Delta) is nef and that, for an effective rational
divisor (D),
[
@@ -6155,7 +6155,7 @@ is surjective.
Proof.
Set (B=Delta-S). The pair ((X,S+B)) is dlt, (B) is an effective
-rational divisor, and (lfloor S+Brfloor=S). Since a dlt pair is log
+rational divisor, and (floor( S+B)=S). Since a dlt pair is log
canonical, Theorem~thm:main makes its nef adjoint semiample.
The pair and the displayed effective representative therefore satisfy the
hypotheses of cite[Proposition~5.12]{FujinoGongyo2014}, which gives the
@@ -6192,7 +6192,7 @@ the source need not be projective over (C).
For an integral Weil divisor (G) on a normal variety (X),
(OO_X(G)) denotes its divisorial sheaf. For a rational divisor (D)
on (X) and nonnegative integers (m,n), the inclusions
-(lfloor mDrfloor+lfloor nDrfloorleqlfloor(m+n)Drfloor)
+(floor( mD)+floor( nD)leqfloor((m+n)D))
give the usual multiplication on the section algebras below.
@@ -6206,7 +6206,7 @@ that (Deltageq0) is rational and (D=K_X+Delta) is (Q)-Cartier.
Then
[
R(X,D)=⊕_{mgeq0}
- H^0(X,OO_X(lfloor mDrfloor))
+ H^0(X,OO_X(floor( mD)))
]
is a finitely generated (k)-algebra.
Let ((X,Delta)) be a log canonical pair over (C), with
@@ -6214,8 +6214,8 @@ is a finitely generated (k)-algebra.
proper morphism (f:Xto S) onto an algebraic variety (S), the relative
log canonical algebra
[
- mathcal R(X/S,K_X+calR(X/S,K_X+Delta)
- =⊕_{mgeq0}f_*OO_X(lfloor m(K_X+Delta)rfloor)
+ R(X/S,K_X+calR(X/S,K_X+Delta)
+ =⊕_{mgeq0}f_*OO_X(floor( m(K_X+Delta)))
]
is a finitely generated (OO_S)-algebra.
@@ -6232,8 +6232,8 @@ We first prove (i). If (D) is not pseudo-effective, a nonzero
section (s) in degree (m>0) would give
[
div(s)+mD
- =(div(s)+lfloor mDrfloor)
- +(mD-lfloor mDrfloor)geq0.
+ =(div(s)+floor( mD))
+ +(mD-floor( mD))geq0.
]
This would make (D) pseudo-effective. Thus all positive graded pieces
vanish in this case. Since (H^0(X,OO_X)=k), we have (R(X,D)=k).
@@ -6259,12 +6259,12 @@ pushed-forward boundary gives equality of these discrepancies. This proves
the asserted support.
For any normal variety (Z), rational Cartier divisor (G), and
-(mgeq0), the nonzero global sections of (OO_Z(lfloor mGrfloor)) are
+(mgeq0), the nonzero global sections of (OO_Z(floor( mG))) are
exactly the rational functions (s) satisfying
[
div_Z(s)+mGgeq0.
]
-This is equivalent to the usual inequality with (lfloor mGrfloor)
+This is equivalent to the usual inequality with (floor( mG))
because the coefficients of (div_Z(s)) are integers.
If the inequality holds on (X), pull back its effective (Q)-Cartier
left side to (W) and push forward by (q). Equation~(eq:ring-comparison)
@@ -6272,8 +6272,8 @@ and (q_*E=0) give the inequality on (Y). Conversely, pull back the
inequality on (Y), add (mE), and push forward by (p). Thus, inside
the common function field,
[
- H^0(X,OO_X(lfloor mDrfloor))
- =H^0(Y,OO_Y(lfloor mD_Yrfloor))
+ H^0(X,OO_X(floor( mD)))
+ =H^0(Y,OO_Y(floor( mD_Y)))
(mgeq0).
]
These identifications preserve multiplication, so they identify the full
@@ -6282,19 +6282,19 @@ graded rings.
It remains to use semiampleness on (Y). Choose (r>0) such that
(rD_Y) is Cartier and globally generated. Its complete linear system
defines (psi:YtoPP^N_k) with
-(OO_Y(rD_Y)simeqpsi^*OO_{PP^N}(1)). Let
+(OO_Y(rD_Y)~=psi^*OO_{PP^N}(1)). Let
(A=k[T_0,...,T_N]) act on (R(Y,D_Y)) by pulling back the coordinate
sections, each in degree (r). For (0leq i<r), put
-(mathcal F_i=calF_i=psi_*OO_Y(lfloor iD_Yrfloor)), which is coherent on
+(F_i=*OO_Y(floor( iD_Y))), which is coherent on
(PP^N_k) because (psi) is proper. Since
[
- lfloor(ell r+i)D_Yrfloor
- =ell(rD_Y)+lfloor iD_Yrfloorqquad(ellgeq0),
+ floor((ell r+i)D_Y)
+ =ell(rD_Y)+floor( iD_Y) (ellgeq0),
]
projection formula identifies the part in degrees congruent to (i)
modulo (r), as an (A)-module, with
[
- ⊕_{ellgeq0}H^0(PP^N_k,mathcal F_i(calF_i(ell)).
+ ⊕_{ellgeq0}H^0(PP^N_k,F_i(
]
This is a finite (A)-module by the standard finite-generation theorem
for section modules cite[Lemma~30.14.1(5)]{StacksSectionModules}.
@@ -6303,7 +6303,7 @@ There are only (r) residue classes, so (R(Y,D_Y)) is finite as an
(i).
For (ii), set (n=dim X). In the notation of Fujino--Gongyo,
-Conjecture (C_{leq n-1}) asks for good minimal models of projective
+Conjecture (C_{<= n-1}) asks for good minimal models of projective
(Q)-factorial dlt pairs with effective real boundary and pseudo-effective
adjoint in dimensions at most (n-1). Theorem~thm:good-models-complex
supplies this statement. Its good-model convention is the one fixed in
@@ -6446,32 +6446,32 @@ MRC base is a point and (X) is rationally connected.
We next use ordinary cotangent invariants, with no boundary. If
-(mathcal LcalL) is a coherent rank-one subsheaf of a vector bundle on a
+(L is a coherent rank-one subsheaf of a vector bundle on a
smooth projective variety, its double dual is a line bundle; put
-(kappa(mathcal L)=calL)=kappa(mathcal L^calL^{**})). Define
+(kappa(L)= L^calL^{**})). Define
[
omega(X)=
max_{substack{m>0
- mathcal LcalLsubset(Ω_X^1)^{ ⊗ m}
- rankmathcal L=1}calL=1}
- kappa(mathcal L),calL),
+ LΩ_X^1)^{ ⊗ m}
+ rankL=1}}
+ kappa(L),
]
where (m) is an integer and the value is (-∞) if no such sheaf
-has nonnegative Iitaka dimension. The inclusion of (mathcal LcalL) extends
+has nonnegative Iitaka dimension. The inclusion of (L extends
to its double dual: away from codimension two these sheaves agree, and a
map into a locally free sheaf extends across that subset. Taking tensor
-powers then realizes the systems (H^0(X,(mathcal L^calL^{**})^{ ⊗ a})),
+powers then realizes the systems (H^0(X,(L^{**})^{ ⊗ a})),
for integers (a>0), inside higher cotangent tensor powers. Thus allowing all (m) makes
this equivalent to Campana's definition using the dimensions of the
-rational maps defined by (H^0(X,mathcal L)calL)). For positive-dimensional (X),
+rational maps defined by (H^0(X,L) For positive-dimensional (X),
also put
[
kappa^+(X)=
- max_{substack{p>00nemathcal FcalFsubsetΩ_X^p}}
- kappa(detmathcal F).calF).
+ max_{substack{p>00neFΩ_X^p}}
+ kappa(detF).
]
-Here (p) is an integer, (mathcal FcalF) is coherent, and
-(detmathcal F=(∧^calF=(∧^{rkmathcal F}calFmathcal F)^calF)^{**}). These are the
+Here (p) is an integer, (F is coherent, and
+(detF=(∧^calF=(∧^{rkF} F)^calF)^{**}). These are the
invariants used in Campana's Appendix~A to Taji [cite].
[Cotangent invariants and the MRC base]
@@ -6569,7 +6569,7 @@ holomorphic fiber bundle (alpha:X'to A) over an abelian variety,
and an 'etale cover (widetilde Ato A) with no positive-dimensional
compact analytic subvarieties, such that
[
- widetilde Xsimeq X'times_Awidetilde A
+ widetilde X~= X'times_Awidetilde A
]
as covering spaces over (X).