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TRANSCRIPT
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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
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 2 / 23
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.
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 3 / 23
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.
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 3 / 23
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.
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 3 / 23
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.
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 3 / 23
Cox ring and Mori dream space
.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.
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 4 / 23
Cox ring and Mori dream space
.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.
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 4 / 23
Cox ring and Mori dream space
.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.
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 4 / 23
Cox ring and Mori dream space
.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.
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 4 / 23
Cox ring and Mori dream space
.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.
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 4 / 23
Cox ring and Mori dream space
.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.
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 4 / 23
Cox ring and Mori dream space
.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.
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 4 / 23
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+,
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 5 / 23
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+,
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 5 / 23
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+,
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 5 / 23
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+,
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 5 / 23
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+,
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 5 / 23
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+,
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 5 / 23
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+,
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 5 / 23
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+,
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 5 / 23
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+,
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 5 / 23
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+,
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 5 / 23
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+,
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 5 / 23
D-minimal model 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 (^),
(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.
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 6 / 23
D-minimal model 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 (^),
(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.
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 6 / 23
D-minimal model 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 (^),
(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.
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 6 / 23
D-minimal model 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 (^),
(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.
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 6 / 23
D-minimal model 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 (^),
(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.
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 6 / 23
D-minimal model 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 (^),
(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.
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 6 / 23
D-minimal model program
.Theorem[Hu–Keel]..
......
X: MDS & D: divisor on X.Then D-MMP run and terminates.Moreover each Xi is also a MDS.
.Theorem[Hu–Keel]..
......
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.
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 7 / 23
D-minimal model program
.Theorem[Hu–Keel]..
......
X: MDS & D: divisor on X.Then D-MMP run and terminates.Moreover each Xi is also a MDS.
.Theorem[Hu–Keel]..
......
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.
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 7 / 23
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.
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 8 / 23
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.
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 8 / 23
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.
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 8 / 23
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.
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 8 / 23
Review of Okawa’s talk
.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.
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 9 / 23
Review of Okawa’s talk
.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.
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 9 / 23
Review of Okawa’s talk
.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.
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 9 / 23
Review of Okawa’s talk
.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.
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 9 / 23
Review of Okawa’s talk
.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.
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 9 / 23
Review of Okawa’s talk
.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.
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 9 / 23
Review of Okawa’s talk
.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.
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 10 / 23
Review of Okawa’s talk
.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.
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 10 / 23
Review of Okawa’s talk
.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.
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 10 / 23
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.
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 11 / 23
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.
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 11 / 23
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.
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 11 / 23
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.
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 11 / 23
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.
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 11 / 23
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.
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 11 / 23
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.
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 11 / 23
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.
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 11 / 23
MDS of gl. F-reg. type is of Fano type
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]..
......
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.
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 12 / 23
MDS of gl. F-reg. type is of Fano type
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]..
......
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.
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 12 / 23
MDS of gl. F-reg. type is of Fano type
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]..
......
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.
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 12 / 23
MDS of gl. F-reg. type is of Fano type
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]..
......
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.
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 12 / 23
MDS of gl. F-reg. type is of Fano type
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]..
......
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.
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 12 / 23
MDS of gl. F-reg. type is of Fano type
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.
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 13 / 23
MDS of gl. F-reg. type is of Fano type
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.
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 13 / 23
MDS of gl. F-reg. type is of Fano type
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.
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 13 / 23
MDS of gl. F-reg. type is of Fano type
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.
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 13 / 23
MDS of gl. F-reg. type is of Fano type
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.
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 13 / 23
MDS of gl. F-reg. type is of Fano type
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.
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 13 / 23
MDS of gl. F-reg. type is of Fano type
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.
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 13 / 23
MDS of gl. F-reg. type is of Fano type
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.
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 13 / 23
MDS of gl. F-reg. type is of Fano type
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.
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 13 / 23
MDS of gl. F-reg. type is of Fano type
In particular, Hara–Watanabe’s theorem says X has only logterminal singularities..Theorem[Hara-Watanabe]..
......
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.
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 14 / 23
MDS of gl. F-reg. type is of Fano type
In particular, Hara–Watanabe’s theorem says X has only logterminal singularities..Theorem[Hara-Watanabe]..
......
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.
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 14 / 23
MDS of gl. F-reg. type is of Fano type
In particular, Hara–Watanabe’s theorem says X has only logterminal singularities..Theorem[Hara-Watanabe]..
......
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.
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 14 / 23
MDS of gl. F-reg. type is of Fano type
In particular, Hara–Watanabe’s theorem says X has only logterminal singularities..Theorem[Hara-Watanabe]..
......
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.
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 14 / 23
MDS of gl. F-reg. type is of Fano type
(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 .
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 15 / 23
MDS of gl. F-reg. type is of Fano type
(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 .
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 15 / 23
MDS of gl. F-reg. type is of Fano type
(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 .
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 15 / 23
MDS of gl. F-reg. type is of Fano type
(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 .
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 15 / 23
MDS of gl. F-reg. type is of Fano type
(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 .
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 15 / 23
MDS of gl. F-reg. type 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.
Finish the proof of Theorem F.
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 16 / 23
MDS of gl. F-reg. type 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.
Finish the proof of Theorem F.
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 16 / 23
MDS of gl. F-reg. type 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.
Finish the proof of Theorem F.
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 16 / 23
MDS of gl. F-reg. type 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.
Finish the proof of Theorem F.
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 16 / 23
MDS of gl. F-reg. type 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.
Finish the proof of Theorem F.
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 16 / 23
MDS of gl. F-reg. type 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.
Finish the proof of Theorem F.
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 16 / 23
MDS of gl. F-reg. type 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.
Finish the proof of Theorem F.
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 16 / 23
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).
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 17 / 23
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).
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 17 / 23
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).
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 17 / 23
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).
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 17 / 23
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).
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 17 / 23
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).
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 17 / 23
Character. of Fano type via Cox rings
(⇒)
.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.
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 18 / 23
Character. of Fano type via Cox rings
(⇒).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.
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 18 / 23
Character. of Fano type via Cox rings
(⇒).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.
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 18 / 23
Character. of Fano type via Cox rings
(⇒).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.
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 18 / 23
Character. of Fano type via Cox rings
(⇒).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.
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 18 / 23
Character. of Fano type via Cox rings
(⇒).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.
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 18 / 23
Character. of Fano type via Cox rings
(⇒).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.
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 18 / 23
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.
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 19 / 23
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.
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 19 / 23
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.
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 19 / 23
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.
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 19 / 23
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.
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 19 / 23
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)
for ∀D ∈ mΓ &∀ closed fibers Xp of some model.
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 20 / 23
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)
for ∀D ∈ mΓ &∀ closed fibers Xp of some model.
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 20 / 23
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)
for ∀D ∈ mΓ &∀ closed fibers Xp of some model.
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 20 / 23
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)
for ∀D ∈ mΓ &∀ closed fibers Xp of some model.
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 20 / 23
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)
for ∀D ∈ mΓ &∀ closed fibers Xp of some model.
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 20 / 23
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)
for ∀D ∈ mΓ &∀ closed fibers Xp of some model.
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 20 / 23
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.
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 21 / 23
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.
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 21 / 23
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.
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 21 / 23
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.
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 21 / 23
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.
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 22 / 23
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.
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 22 / 23
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.
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 22 / 23
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.
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 22 / 23
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.
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 22 / 23
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.
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 23 / 23
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.
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 23 / 23
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.
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 23 / 23