Characterization of varieties of Fano type via ...cb496/conf-chula/talk-gongyo.pdf5 Characterization of varieties of Fano type via singularities of Cox rings 6 Proof of Key lemma 7

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Characterization of varieties of Fano type viasingularities of Cox rings II

Y. Gongyo, S. Okawa, A. Sannai, & S. TakagiUniversity of Tokyo

Algebraic Geometry ConferenceChulalongkorn University, Bangkok, Thailand

21, December, 2011

Y. Gongyo, S. Okawa, A. Sannai, & S. Takagi University of Tokyo Algebraic Geometry Conference Chulalongkorn University, Bangkok, Thailand ()Character. of Fano type via Cox rings II 21, December, 2011 1 / 23

Schedule

1 Cox ring and Mori dream space

2 D-minimal model program

3 Review of Okawa’s talk

4 MDS of gl. F-reg. type is of Fano type

5 Characterization of varieties of Fano type via singularities of Coxrings

6 Proof of Key lemma

7 Case of log Calabi–Yau

8 Application

9 Open question


Cox ring and Mori dream space

.Definition(Cox ring)..

......

X: a normal Q-fac. proj. var. / k

s.t. Pic (X)Q ≃ N1(X)Q.Γ ⊂ Div (X): f.g. group of Cartier div. on X s.t.

ΓQ → Pic (X)Q; D 7→ OX(D)

is iso. The multi-sec. ring

RX(Γ) =⊕D∈Γ

H0(X,OX(D))

is called a Cox ring of X.




......

X: a normal Q-fac. proj. var. / k s.t. Pic (X)Q ≃ N1(X)Q.

Γ ⊂ Div (X): f.g. group of Cartier div. on X s.t.



RX(Γ) =⊕D∈Γ

H0(X,OX(D))





......

X: a normal Q-fac. proj. var. / k s.t. Pic (X)Q ≃ N1(X)Q.Γ ⊂ Div (X): f.g. group of Cartier div. on X s.t.


is iso.

The multi-sec. ring

RX(Γ) =⊕D∈Γ

H0(X,OX(D))





......

X: a normal Q-fac. proj. var. / k s.t. Pic (X)Q ≃ N1(X)Q.Γ ⊂ Div (X): f.g. group of Cartier div. on X s.t.



RX(Γ) =⊕D∈Γ

H0(X,OX(D))




.Definition[Mori dream space]..

......

X: normal proj. var. / k.X : Mori dream space (or MDS for short)⇔

(i) X : Q-fac. & Pic(X)Q ≃ N1(X)Q,

(ii) Nef(X) is the affine hull of finitely many semi-ample linebundles,

(iii) ∃ a finite collection of small bir. maps fi : X d Xi s.t. Xi

satisfies (i) & (ii), and Mov(X) =∪

i f ∗i(Nef(Xi)).

.Theorem[Hu–Keel]..

......

X: Q-fac. normal proj. var. / k such that Pic(X)Q ≃ N1(X)Q.X : MDS⇔ Cox(X) : f.g. k-alg.




......

X: normal proj. var. / k.

X : Mori dream space (or MDS for short)⇔(i) X : Q-fac. & Pic(X)Q ≃ N1(X)Q,




i f ∗i(Nef(Xi)).


......





......

X: normal proj. var. / k.X : Mori dream space (or MDS for short)

⇔(i) X : Q-fac. & Pic(X)Q ≃ N1(X)Q,




i f ∗i(Nef(Xi)).


......





......






i f ∗i(Nef(Xi)).


......





......






i f ∗i(Nef(Xi)).


......





......






i f ∗i(Nef(Xi)).


......





......






i f ∗i(Nef(Xi)).


......



D-minimal model program

.D-MMP..

......

Given X & D − (^),Ask D is nef or not,D is nef⇒ X: D-minimal model,D is not nef⇒ Find a curve C s.t. D.C < 0 & R := R≥0[C] ⊂ N1(X)R: extremal,⇒ Construct a proj. mor. φ : X → Y of conn. fibers s.t.

Curve C′ s.t. φ(C) is a point⇔ [C′] ∈ R,

(i) φ: bir. contracts a divisor⇒ Replacing as X := Y & D := φ∗D, go back (^),

(ii) φ: bir. contracts no divisors,⇒ Constructing D-flip X d X+,



.D-MMP..

......

Given X & D − (^),

Ask D is nef or not,D is nef⇒ X: D-minimal model,D is not nef⇒ Find a curve C s.t. D.C < 0 & R := R≥0[C] ⊂ N1(X)R: extremal,⇒ Construct a proj. mor. φ : X → Y of conn. fibers s.t.






.D-MMP..

......

Given X & D − (^),Ask D is nef or not,

D is nef⇒ X: D-minimal model,D is not nef⇒ Find a curve C s.t. D.C < 0 & R := R≥0[C] ⊂ N1(X)R: extremal,⇒ Construct a proj. mor. φ : X → Y of conn. fibers s.t.






.D-MMP..

......

Given X & D − (^),Ask D is nef or not,D is nef⇒ X: D-minimal model,

D is not nef⇒ Find a curve C s.t. D.C < 0 & R := R≥0[C] ⊂ N1(X)R: extremal,⇒ Construct a proj. mor. φ : X → Y of conn. fibers s.t.






.D-MMP..

......

Given X & D − (^),Ask D is nef or not,D is nef⇒ X: D-minimal model,D is not nef

⇒ Find a curve C s.t. D.C < 0 & R := R≥0[C] ⊂ N1(X)R: extremal,⇒ Construct a proj. mor. φ : X → Y of conn. fibers s.t.






.D-MMP..

......

Given X & D − (^),Ask D is nef or not,D is nef⇒ X: D-minimal model,D is not nef⇒ Find a curve C s.t. D.C < 0 & R := R≥0[C] ⊂ N1(X)R: extremal,

⇒ Construct a proj. mor. φ : X → Y of conn. fibers s.t.






.D-MMP..

......







.D-MMP..

......



(i) φ: bir. contracts a divisor

⇒ Replacing as X := Y & D := φ∗D, go back (^),




.D-MMP..

......







.D-MMP..

......




(ii) φ: bir. contracts no divisors,

⇒ Constructing D-flip X d X+,



.D-MMP..

......







.D-MMP..

......

i.e. ∃ small bir. mor. φ+ : X+ → Y s.t. X+: Q-fac., ρ(X+/Y) = 1, &the str. trans. D+: φ+-ample,

⇒ Replacing as X := X+ & D := D+, go back (^),

(iii) φ has a positive dim. general fibers.⇒ φ : X → Y: D- Mori fiber space.

Repeat the process:

X0 = X d X1 d · · · d Xi · · · .

For looking for D-minimal models or D-Mori fiber spaces, we runthe program.



.D-MMP..

......

i.e. ∃ small bir. mor. φ+ : X+ → Y s.t. X+: Q-fac., ρ(X+/Y) = 1, &the str. trans. D+: φ+-ample,⇒ Replacing as X := X+ & D := D+, go back (^),


Repeat the process:

X0 = X d X1 d · · · d Xi · · · .




.D-MMP..

......


(iii) φ has a positive dim. general fibers.

⇒ φ : X → Y: D- Mori fiber space.

Repeat the process:

X0 = X d X1 d · · · d Xi · · · .




.D-MMP..

......



Repeat the process:

X0 = X d X1 d · · · d Xi · · · .




.D-MMP..

......



Repeat the process:

X0 = X d X1 d · · · d Xi · · · .




.D-MMP..

......



Repeat the process:

X0 = X d X1 d · · · d Xi · · · .





......

X: MDS & D: divisor on X.Then D-MMP run and terminates.Moreover each Xi is also a MDS.


......

X: MDS.Then ∃ a finite collection of rat. cont. maps fi : X d Xi s.t. ∀ rat.cont. map g : X d Y, ∃i s.t. fi ≃ g.




......

X: MDS & D: divisor on X.Then D-MMP run and terminates.Moreover each Xi is also a MDS.


......

X: MDS.Then ∃ a finite collection of rat. cont. maps fi : X d Xi s.t. ∀ rat.cont. map g : X d Y, ∃i s.t. fi ≃ g.


Review of Okawa’s talk

Review of Okawa’s talk.Strongly F-regular..

......

k: F-finite field of char. p > 0.An int. dom. R is f.g. alg./ k isR: strongly F-regular⇔ 0 , ∀c ∈ R∃e > 0 s.t.

cFe : R → Fe∗R → Fe

∗R

splits as R-mod.




......

k: F-finite field of char. p > 0.

An int. dom. R is f.g. alg./ k isR: strongly F-regular⇔ 0 , ∀c ∈ R∃e > 0 s.t.


∗R

splits as R-mod.




......

k: F-finite field of char. p > 0.An int. dom. R is f.g. alg./ k isR: strongly F-regular

⇔ 0 , ∀c ∈ R∃e > 0 s.t.


∗R

splits as R-mod.




......

k: F-finite field of char. p > 0.An int. dom. R is f.g. alg./ k isR: strongly F-regular⇔ 0 , ∀c ∈ R∃e > 0 s.t.


∗R

splits as R-mod.



.Globally F-regular..

......

k: F-finite of char. p > 0.X: normal var./ k.

X : globally F-regular⇔ ∀D ≥ 0∃e > 0 s.t.

cFe : OX → Fe∗OX → Fe

∗(OX(D)),

splits as OX-mod. , where c ∈ H0(X,OX(D)) s.t. div0(c) = D.

.Proposition (K. Smith)..

......

As above,X is gl. F-regular⇔ ∃ ample div. H s.t. the sec. ring R(X,H):strongly F-regular.




......

k: F-finite of char. p > 0.X: normal var./ k.X : globally F-regular

⇔ ∀D ≥ 0∃e > 0 s.t.


∗(OX(D)),



......





......

k: F-finite of char. p > 0.X: normal var./ k.X : globally F-regular⇔ ∀D ≥ 0∃e > 0 s.t.


∗(OX(D)),



......





......



∗(OX(D)),



......

As above,

X is gl. F-regular⇔ ∃ ample div. H s.t. the sec. ring R(X,H):strongly F-regular.




......



∗(OX(D)),



......

As above,X is gl. F-regular

⇔ ∃ ample div. H s.t. the sec. ring R(X,H):strongly F-regular.




......



∗(OX(D)),



......




.Theorem P (Hashimoto, Sannai)..

......

k: F-finite of char. p > 0.X: proj. normal var./ k.

X: gl. F-regular.⇔ ∀ Γ ⊂ Div (X): semi-group of Cartier div.

RX(Γ) =⊕D∈Γ

H0(X,OX(D))

is strongly F-regular.




......

k: F-finite of char. p > 0.X: proj. normal var./ k.X: gl. F-regular.

⇔ ∀ Γ ⊂ Div (X): semi-group of Cartier div.

RX(Γ) =⊕D∈Γ

H0(X,OX(D))





......

k: F-finite of char. p > 0.X: proj. normal var./ k.X: gl. F-regular.⇔ ∀ Γ ⊂ Div (X): semi-group of Cartier div.

RX(Γ) =⊕D∈Γ

H0(X,OX(D))



MDS of gl. F-reg. type is of Fano type

.Theorem F [GOST]..

......

X: MDS.

If X is of gl. F-regular type,then X is of Fano type, i.e.,∃∆ ≥ 0 s.t. (X,∆) is klt & −(KX + ∆): ample.

Proof. Running (−KX)-MMP:

X0 = X d X1 d · · · d Xl,

where Xl: a final model. We know each Xi is also a MDS of gl.F-reg. type since, in general,any images of ver. of gl. F-reg. type are also of gl. F-reg. type,and gl. F-reg. preserve under isom. in codim 1.



.Theorem F [GOST]..

......

X: MDS.If X is of gl. F-regular type,

then X is of Fano type, i.e.,∃∆ ≥ 0 s.t. (X,∆) is klt & −(KX + ∆): ample.


X0 = X d X1 d · · · d Xl,




.Theorem F [GOST]..

......

X: MDS.If X is of gl. F-regular type,then X is of Fano type, i.e.,∃∆ ≥ 0 s.t. (X,∆) is klt & −(KX + ∆): ample.


X0 = X d X1 d · · · d Xl,




.Theorem F [GOST]..

......


Proof.

Running (−KX)-MMP:

X0 = X d X1 d · · · d Xl,




.Theorem F [GOST]..

......



X0 = X d X1 d · · · d Xl,




.Theorem F [GOST]..

......



X0 = X d X1 d · · · d Xl,




.Theorem F [GOST]..

......



X0 = X d X1 d · · · d Xl,

where Xl: a final model.

We know each Xi is also a MDS of gl.F-reg. type since, in general,any images of ver. of gl. F-reg. type are also of gl. F-reg. type,and gl. F-reg. preserve under isom. in codim 1.



.Theorem F [GOST]..

......



X0 = X d X1 d · · · d Xl,




Thus we see:

.Claim........Xl is (−KX)-minimal model such that −KXl is big.

Proof of Claim: We know Xl,p is of Fano type under taking mod.p-reduction from Schwede–Smith’s theorem:.Theorem[Schwede–Smith]..

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Xp : gl. F-regular variety over a F-finite field of characteristicp > 0.Then Xp is of Fano type, i.e. ∃∆p ≥ 0 s.t. (Xp,∆p) is klt &−(KXp + ∆p): ample.

Remark! ∆p depends on p. Thus we can not lift it on X.



Thus we see:.Claim........Xl is (−KX)-minimal model such that −KXl is big.


......






Proof of Claim:

We know Xl,p is of Fano type under taking mod.p-reduction from Schwede–Smith’s theorem:.Theorem[Schwede–Smith]..

......







......







......





Assume f : Xl → Y is (−KX)-Mori fiber space.

Let C af -contracting curve.

KXl .C > 0.

Taking reduction mod p, it holds KXl,p .Cp > 0 and Cp is movable.However ∃∆p ≥ 0 s.t. & −(KXp + ∆p): ample.Since ∆p.Cp ≥ 0, this is a contradiction!Thus −KXl is nef. In particular, −KXl is semi-ample since Xl is aMDS. Thus we see:

(−KXl)dim X = (−KXl,p)dim X > 0,

since −KXl,p is nef and big. Finish the proof of Claim.



Assume f : Xl → Y is (−KX)-Mori fiber space. Let C af -contracting curve.

KXl .C > 0.







KXl .C > 0.

Taking reduction mod p,

it holds KXl,p .Cp > 0 and Cp is movable.However ∃∆p ≥ 0 s.t. & −(KXp + ∆p): ample.Since ∆p.Cp ≥ 0, this is a contradiction!Thus −KXl is nef. In particular, −KXl is semi-ample since Xl is aMDS. Thus we see:






KXl .C > 0.

Taking reduction mod p, it holds KXl,p .Cp > 0 and Cp is movable.

However ∃∆p ≥ 0 s.t. & −(KXp + ∆p): ample.Since ∆p.Cp ≥ 0, this is a contradiction!Thus −KXl is nef. In particular, −KXl is semi-ample since Xl is aMDS. Thus we see:






KXl .C > 0.

Taking reduction mod p, it holds KXl,p .Cp > 0 and Cp is movable.However ∃∆p ≥ 0 s.t. & −(KXp + ∆p): ample.

Since ∆p.Cp ≥ 0, this is a contradiction!Thus −KXl is nef. In particular, −KXl is semi-ample since Xl is aMDS. Thus we see:






KXl .C > 0.

Taking reduction mod p, it holds KXl,p .Cp > 0 and Cp is movable.However ∃∆p ≥ 0 s.t. & −(KXp + ∆p): ample.Since ∆p.Cp ≥ 0, this is a contradiction!

Thus −KXl is nef. In particular, −KXl is semi-ample since Xl is aMDS. Thus we see:






KXl .C > 0.

Taking reduction mod p, it holds KXl,p .Cp > 0 and Cp is movable.However ∃∆p ≥ 0 s.t. & −(KXp + ∆p): ample.Since ∆p.Cp ≥ 0, this is a contradiction!Thus −KXl is nef. In particular, −KXl is semi-ample since Xl is aMDS.

Thus we see:






KXl .C > 0.



since −KXl,p is nef and big.

Finish the proof of Claim.




KXl .C > 0.






In particular, Hara–Watanabe’s theorem says X has only logterminal singularities..Theorem[Hara-Watanabe]..

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X: Q-Gor. normal var./ k of ch.=0.If X is of strongly F-regular type, then X has only log terminalsingularities.

Thus Xl is of Fano type.Now by induction it suffice to show X is of Fano type under theassumption that X1 is of Fano type.Assume ∃∆1 ≥ 0 s.t. (X1,∆1) is klt & −(KX1 + ∆1): ample.




......


Thus Xl is of Fano type.

Now by induction it suffice to show X is of Fano type under theassumption that X1 is of Fano type.Assume ∃∆1 ≥ 0 s.t. (X1,∆1) is klt & −(KX1 + ∆1): ample.




......


Thus Xl is of Fano type.Now by induction it suffice to show X is of Fano type under theassumption that X1 is of Fano type.

Assume ∃∆1 ≥ 0 s.t. (X1,∆1) is klt & −(KX1 + ∆1): ample.




......


Thus Xl is of Fano type.Now by induction it suffice to show X is of Fano type under theassumption that X1 is of Fano type.Assume ∃∆1 ≥ 0 s.t. (X1,∆1) is klt & −(KX1 + ∆1): ample.



(i) When f : X d X1: flip,

let g : X → Y be the flipping contr. &g+ : X1 → Y the flipped contr.⇒ Y is of Fano type since, in general, images of var. of Fanotype are also of Fano type,i.e.∃∆Y ≥ 0 s.t. (Y,∆Y) is klt & −(KY + ∆Y): ample.⇒ X is of Fano type since −(KX + ∆) = −g+∗(KY + ∆Y) isnef-big, and klt, where ∆: the str. trans. of ∆Y .



(i) When f : X d X1: flip,let g : X → Y be the flipping contr. &g+ : X1 → Y the flipped contr.

⇒ Y is of Fano type since, in general, images of var. of Fanotype are also of Fano type,i.e.∃∆Y ≥ 0 s.t. (Y,∆Y) is klt & −(KY + ∆Y): ample.⇒ X is of Fano type since −(KX + ∆) = −g+∗(KY + ∆Y) isnef-big, and klt, where ∆: the str. trans. of ∆Y .



(i) When f : X d X1: flip,let g : X → Y be the flipping contr. &g+ : X1 → Y the flipped contr.⇒ Y is of Fano type since, in general, images of var. of Fanotype are also of Fano type,

i.e.∃∆Y ≥ 0 s.t. (Y,∆Y) is klt & −(KY + ∆Y): ample.⇒ X is of Fano type since −(KX + ∆) = −g+∗(KY + ∆Y) isnef-big, and klt, where ∆: the str. trans. of ∆Y .



(i) When f : X d X1: flip,let g : X → Y be the flipping contr. &g+ : X1 → Y the flipped contr.⇒ Y is of Fano type since, in general, images of var. of Fanotype are also of Fano type,i.e.∃∆Y ≥ 0 s.t. (Y,∆Y) is klt & −(KY + ∆Y): ample.

⇒ X is of Fano type since −(KX + ∆) = −g+∗(KY + ∆Y) isnef-big, and klt, where ∆: the str. trans. of ∆Y .



(i) When f : X d X1: flip,let g : X → Y be the flipping contr. &g+ : X1 → Y the flipped contr.⇒ Y is of Fano type since, in general, images of var. of Fanotype are also of Fano type,i.e.∃∆Y ≥ 0 s.t. (Y,∆Y) is klt & −(KY + ∆Y): ample.⇒ X is of Fano type since −(KX + ∆) = −g+∗(KY + ∆Y) isnef-big, and klt, where ∆: the str. trans. of ∆Y .



(ii) When f : X d X1: divisorial contr.,

we know KX is f -ample.Let ∆: the str. trans. of ∆1.⇒ ∆ is f -nef, in particular, KX + ∆: f -ample,⇒ −(KX + ∆) = − f ∗(KX1 + ∆1) + aE,where E the f -excep. div. and a > 0 ( from the negativitylemma).Thus (X,∆ + aE) is klt and −(KX + ∆ + aE) is nef-big. Inparticular, X is of Fano type.

Finish the proof of Theorem F.



(ii) When f : X d X1: divisorial contr., we know KX is f -ample.

Let ∆: the str. trans. of ∆1.⇒ ∆ is f -nef, in particular, KX + ∆: f -ample,⇒ −(KX + ∆) = − f ∗(KX1 + ∆1) + aE,where E the f -excep. div. and a > 0 ( from the negativitylemma).Thus (X,∆ + aE) is klt and −(KX + ∆ + aE) is nef-big. Inparticular, X is of Fano type.




(ii) When f : X d X1: divisorial contr., we know KX is f -ample.Let ∆: the str. trans. of ∆1.⇒ ∆ is f -nef

, in particular, KX + ∆: f -ample,⇒ −(KX + ∆) = − f ∗(KX1 + ∆1) + aE,where E the f -excep. div. and a > 0 ( from the negativitylemma).Thus (X,∆ + aE) is klt and −(KX + ∆ + aE) is nef-big. Inparticular, X is of Fano type.




(ii) When f : X d X1: divisorial contr., we know KX is f -ample.Let ∆: the str. trans. of ∆1.⇒ ∆ is f -nef, in particular, KX + ∆: f -ample,

⇒ −(KX + ∆) = − f ∗(KX1 + ∆1) + aE,where E the f -excep. div. and a > 0 ( from the negativitylemma).Thus (X,∆ + aE) is klt and −(KX + ∆ + aE) is nef-big. Inparticular, X is of Fano type.




(ii) When f : X d X1: divisorial contr., we know KX is f -ample.Let ∆: the str. trans. of ∆1.⇒ ∆ is f -nef, in particular, KX + ∆: f -ample,⇒ −(KX + ∆) = − f ∗(KX1 + ∆1) + aE,where E the f -excep. div. and a > 0 ( from the negativitylemma).

Thus (X,∆ + aE) is klt and −(KX + ∆ + aE) is nef-big. Inparticular, X is of Fano type.




(ii) When f : X d X1: divisorial contr., we know KX is f -ample.Let ∆: the str. trans. of ∆1.⇒ ∆ is f -nef, in particular, KX + ∆: f -ample,⇒ −(KX + ∆) = − f ∗(KX1 + ∆1) + aE,where E the f -excep. div. and a > 0 ( from the negativitylemma).Thus (X,∆ + aE) is klt and −(KX + ∆ + aE) is nef-big. Inparticular, X is of Fano type.




(ii) When f : X d X1: divisorial contr., we know KX is f -ample.Let ∆: the str. trans. of ∆1.⇒ ∆ is f -nef, in particular, KX + ∆: f -ample,⇒ −(KX + ∆) = − f ∗(KX1 + ∆1) + aE,where E the f -excep. div. and a > 0 ( from the negativitylemma).Thus (X,∆ + aE) is klt and −(KX + ∆ + aE) is nef-big. Inparticular, X is of Fano type.



Character. of Fano type via Cox rings

.Theorem C [GOST]..

......

Let X be a Q-fac. proj. normal var./ C.X : of Fano type⇔ Cox(X) has only log terminal singularities.

Proof. (⇐)Let H be a very ample divisor on X.From Theorem F, it suffices to show R(X,H) is strongly F-regulartype.That follows from the fact that R(X,H) is a pure subring of Cox(X)after even taking a mod p-reduction.Remark that we don’t know Cox(X)p = Cox(Xp).



.Theorem C [GOST]..

......


Proof. (⇐)

Let H be a very ample divisor on X.From Theorem F, it suffices to show R(X,H) is strongly F-regulartype.That follows from the fact that R(X,H) is a pure subring of Cox(X)after even taking a mod p-reduction.Remark that we don’t know Cox(X)p = Cox(Xp).



.Theorem C [GOST]..

......


Proof. (⇐)Let H be a very ample divisor on X.

From Theorem F, it suffices to show R(X,H) is strongly F-regulartype.That follows from the fact that R(X,H) is a pure subring of Cox(X)after even taking a mod p-reduction.Remark that we don’t know Cox(X)p = Cox(Xp).



.Theorem C [GOST]..

......


Proof. (⇐)Let H be a very ample divisor on X.From Theorem F, it suffices to show R(X,H) is strongly F-regulartype.

That follows from the fact that R(X,H) is a pure subring of Cox(X)after even taking a mod p-reduction.Remark that we don’t know Cox(X)p = Cox(Xp).



.Theorem C [GOST]..

......


Proof. (⇐)Let H be a very ample divisor on X.From Theorem F, it suffices to show R(X,H) is strongly F-regulartype.That follows from the fact that R(X,H) is a pure subring of Cox(X)after even taking a mod p-reduction.

Remark that we don’t know Cox(X)p = Cox(Xp).



.Theorem C [GOST]..

......


Proof. (⇐)Let H be a very ample divisor on X.From Theorem F, it suffices to show R(X,H) is strongly F-regulartype.That follows from the fact that R(X,H) is a pure subring of Cox(X)after even taking a mod p-reduction.Remark that we don’t know Cox(X)p = Cox(Xp).



(⇒)

.Key lemma..

......

X: MDS of gl. F-reg. type / C & Γ: f.g. semi-group of Cartierdivisors.Then there exists m ∈ N s.t. R(X,mΓ)p = R(Xp,mΓp) for a mod preduction.

Cox(X) := R(X, Γ).X: MDS of gl. F-reg. type (from BCHM and Schwede–Smith )⇒ R(X,mΓ)p = R(Xp,mΓp) is strong. F-reg. (Theorem P),⇒ R(X,mΓ) is log terminal (from Hara–Watanabe),⇒ R(X, Γ) is log terminalsince R(X,mΓ) ⊆ R(X, Γ) is etale in codim. 1 (cf. in Okawa’s talk).Finish the proof of Theorem C.



(⇒).Key lemma..

......





(⇒).Key lemma..

......


Cox(X) := R(X, Γ).X: MDS of gl. F-reg. type (from BCHM and Schwede–Smith )

⇒ R(X,mΓ)p = R(Xp,mΓp) is strong. F-reg. (Theorem P),⇒ R(X,mΓ) is log terminal (from Hara–Watanabe),⇒ R(X, Γ) is log terminalsince R(X,mΓ) ⊆ R(X, Γ) is etale in codim. 1 (cf. in Okawa’s talk).Finish the proof of Theorem C.



(⇒).Key lemma..

......


Cox(X) := R(X, Γ).X: MDS of gl. F-reg. type (from BCHM and Schwede–Smith )⇒ R(X,mΓ)p = R(Xp,mΓp) is strong. F-reg. (Theorem P),

⇒ R(X,mΓ) is log terminal (from Hara–Watanabe),⇒ R(X, Γ) is log terminalsince R(X,mΓ) ⊆ R(X, Γ) is etale in codim. 1 (cf. in Okawa’s talk).Finish the proof of Theorem C.



(⇒).Key lemma..

......


Cox(X) := R(X, Γ).X: MDS of gl. F-reg. type (from BCHM and Schwede–Smith )⇒ R(X,mΓ)p = R(Xp,mΓp) is strong. F-reg. (Theorem P),⇒ R(X,mΓ) is log terminal (from Hara–Watanabe),

⇒ R(X, Γ) is log terminalsince R(X,mΓ) ⊆ R(X, Γ) is etale in codim. 1 (cf. in Okawa’s talk).Finish the proof of Theorem C.



(⇒).Key lemma..

......


Cox(X) := R(X, Γ).X: MDS of gl. F-reg. type (from BCHM and Schwede–Smith )⇒ R(X,mΓ)p = R(Xp,mΓp) is strong. F-reg. (Theorem P),⇒ R(X,mΓ) is log terminal (from Hara–Watanabe),⇒ R(X, Γ) is log terminalsince R(X,mΓ) ⊆ R(X, Γ) is etale in codim. 1 (cf. in Okawa’s talk).

Finish the proof of Theorem C.



(⇒).Key lemma..

......




Proof of Key lemma

We give the proof of Key lemma.

Show ∃ m ∈ N s.t.

H0(X,D) = H0(Xp,Dp)

for ∀D ∈ mΓ &∀ closed fibers Xp of some model.∃ a finite collection of birat. cont. maps fi : X d Xi & cont. mor.gi, j : Xi → Yi, j s.t. ∀D ∈ Γ, ∃i, j s.t. fi : X d Xi is some D-MMP withgi, j D-can. model or D-Mori fiber space.

(i) When D is not effective, we see

H0(X,D) = H0(Xp,Dp) = 0

as a similar arguments to the proof of Claim.


Proof of Key lemma

We give the proof of Key lemma.Show ∃ m ∈ N s.t.

H0(X,D) = H0(Xp,Dp)

for ∀D ∈ mΓ &∀ closed fibers Xp of some model.

∃ a finite collection of birat. cont. maps fi : X d Xi & cont. mor.gi, j : Xi → Yi, j s.t. ∀D ∈ Γ, ∃i, j s.t. fi : X d Xi is some D-MMP withgi, j D-can. model or D-Mori fiber space.


H0(X,D) = H0(Xp,Dp) = 0



Proof of Key lemma


H0(X,D) = H0(Xp,Dp)



H0(X,D) = H0(Xp,Dp) = 0



Proof of Key lemma


H0(X,D) = H0(Xp,Dp)


(i) When D is not effective,

we see

H0(X,D) = H0(Xp,Dp) = 0



Proof of Key lemma


H0(X,D) = H0(Xp,Dp)



H0(X,D) = H0(Xp,Dp) = 0



Proof of Key lemma

(ii) When D is effective,

we see

H0(X,D) = H0(Xi, fi∗D).

We can take m ∈ N (do not depend on D, i, j) such thatfi∗mD = g∗

i, jH for some ample div. H on Yi, j.

By the base change theorem and a vanishing theorem forF-split var.,

H0(Yi, j,A) = H0(Yi, j,p,Ap)

for ∀ closed fibers Yi, j,p of any model.

Thus we see ∃ m ∈ N s.t.

H0(X,D) = H0(Xp,Dp)



Proof of Key lemma

(ii) When D is effective, we see

H0(X,D) = H0(Xi, fi∗D).







H0(X,D) = H0(Xp,Dp)



Proof of Key lemma


H0(X,D) = H0(Xi, fi∗D).







H0(X,D) = H0(Xp,Dp)



Proof of Key lemma


H0(X,D) = H0(Xi, fi∗D).







H0(X,D) = H0(Xp,Dp)



Proof of Key lemma


H0(X,D) = H0(Xi, fi∗D).







H0(X,D) = H0(Xp,Dp)



Proof of Key lemma


H0(X,D) = H0(Xi, fi∗D).







H0(X,D) = H0(Xp,Dp)



Case of log Calabi–Yau

.Theorem CY [GOST]..

......

X: MDS.

If X is of dense gl. F-split type, then X is of Calabi–Yau type, i.e.,∃∆ ≥ 0 s.t. (X,∆) is lc & KX + ∆ ∼Q 0.

.Theorem +[Fujino–Takagi]..

......

X: klt Mori dream surface such that KX ∼Q 0.Then Cox(X) has only lc singularities.




......

X: MDS.If X is of dense gl. F-split type, then X is of Calabi–Yau type, i.e.,∃∆ ≥ 0 s.t. (X,∆) is lc & KX + ∆ ∼Q 0.


......





......



......

X: klt Mori dream surface such that KX ∼Q 0.

Then Cox(X) has only lc singularities.




......



......



Application

We can give another proof of the following:.Cor. [Shokurov– Prokhorov, Fujino–Gongyo]..

......

f : X → Y: surj. prom. mor. of normal proj. var. /C.If X is of Fano type then so is Y.

Proof:For simplicity, assume that f has connected fibers.Then Y is also a MDS of gl. F-reg. type (cf. Okawa, ”On images ofMori dream spaces”).By Theorem F, Y is of Fano type.


Application


......


Proof:

For simplicity, assume that f has connected fibers.Then Y is also a MDS of gl. F-reg. type (cf. Okawa, ”On images ofMori dream spaces”).By Theorem F, Y is of Fano type.


Application


......


Proof:For simplicity, assume that f has connected fibers.

Then Y is also a MDS of gl. F-reg. type (cf. Okawa, ”On images ofMori dream spaces”).By Theorem F, Y is of Fano type.


Application


......


Proof:For simplicity, assume that f has connected fibers.Then Y is also a MDS of gl. F-reg. type (cf. Okawa, ”On images ofMori dream spaces”).

By Theorem F, Y is of Fano type.


Application


......


Proof:For simplicity, assume that f has connected fibers.Then Y is also a MDS of gl. F-reg. type (cf. Okawa, ”On images ofMori dream spaces”).By Theorem F, Y is of Fano type.


Open question

.Question 1..

......

X: MDS.Then X is of CY type if and only if Cox(X) has only lc sing.

.Question 2 (Schwede–Smith)........If X is of gl. F-reg. type then X is MDS.

.Question 3 (Schwede–Smith)........X is of dense gl. F-split type if and only if X is of CY type.


Open question

.Question 1..

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Open question

.Question 1..

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