## 代数幾何学セミナー

過去の記録 ～05/23｜次回の予定｜今後の予定 05/24～

開催情報 | 火曜日 15:30～17:00 数理科学研究科棟(駒場) 122号室 |
---|---|

担当者 | 權業 善範・中村 勇哉・田中公 |

**過去の記録**

### 2018年05月21日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

今週は月曜日にセミナーを行います。13:30-15:00と15:30-17:00の2講演あります。This week's seminar will be held on Monday and consist of two lectures: 13:30-15:00 and 15:30-17:00.

Towards the termination of flips. (English)

https://www.math.utah.edu/~hacon/

今週は月曜日にセミナーを行います。13:30-15:00と15:30-17:00の2講演あります。This week's seminar will be held on Monday and consist of two lectures: 13:30-15:00 and 15:30-17:00.

**Christopher Hacon 氏**(Utah/Kyoto)Towards the termination of flips. (English)

[ 講演概要 ]

The minimal model program (MMP) predicts that if $X$ is a smooth complex projective variety which is not uniruled, then there is a finite sequence of "elementary" birational maps

$X=X_0-->X_1-->X_2-->...-->X_n$ known as divisorial contractions and flips whose output $\bar X=X_n$ is a minimal model so that $K_{\bar X}$ is a nef $Q$-divisor i.e it intersects all curves $C\subset \bar X$ non-negatively: $K_{\bar X}\cdot C\geq 0$.

The existence of these birational maps has been established, but in order to complete the MMP, it is necessary to show that flips terminate i.e. there are no infinite sequences of flips. In this talk we will discuss recent results towards the termination of flips.

[ 講演参考URL ]The minimal model program (MMP) predicts that if $X$ is a smooth complex projective variety which is not uniruled, then there is a finite sequence of "elementary" birational maps

$X=X_0-->X_1-->X_2-->...-->X_n$ known as divisorial contractions and flips whose output $\bar X=X_n$ is a minimal model so that $K_{\bar X}$ is a nef $Q$-divisor i.e it intersects all curves $C\subset \bar X$ non-negatively: $K_{\bar X}\cdot C\geq 0$.

The existence of these birational maps has been established, but in order to complete the MMP, it is necessary to show that flips terminate i.e. there are no infinite sequences of flips. In this talk we will discuss recent results towards the termination of flips.

https://www.math.utah.edu/~hacon/

### 2018年05月21日(月)

13:30-15:00 数理科学研究科棟(駒場) 122号室

今週は月曜日にセミナーを行います。13:30-15:00と15:30-17:00の2講演あります。This week's seminar will be held on Monday and consist of two lectures: 13:30-15:00 and 15:30-17:00.

Perverse sheaves of categories and birational geometry (English)

今週は月曜日にセミナーを行います。13:30-15:00と15:30-17:00の2講演あります。This week's seminar will be held on Monday and consist of two lectures: 13:30-15:00 and 15:30-17:00.

**Will Donovan 氏**(IPMU)Perverse sheaves of categories and birational geometry (English)

[ 講演概要 ]

Kapranov and Schechtman have initiated a program to study perverse sheaves of categories, or perverse schobers. It is expected that examples arise from birational geometry, in particular from webs of flops. I explain progress towards constructing these objects for Grothendieck resolutions (work of the above authors with Bondal), and for 3-folds (joint work of myself and Wemyss).

Kapranov and Schechtman have initiated a program to study perverse sheaves of categories, or perverse schobers. It is expected that examples arise from birational geometry, in particular from webs of flops. I explain progress towards constructing these objects for Grothendieck resolutions (work of the above authors with Bondal), and for 3-folds (joint work of myself and Wemyss).

### 2018年05月08日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Higher order families of lines and Fano manifolds covered by linear

spaces

(Japanese (writing in English))

**鈴木拓 氏**(宇都宮大)Higher order families of lines and Fano manifolds covered by linear

spaces

(Japanese (writing in English))

[ 講演概要 ]

In this talk, for an embedded Fano manifold $X$, we introduce higher

order families of lines and a new invariant $S_X$. They are line

versions of higher order minimal families of rational curves and the

invariant $N_X$ which were introduced in my previous talk on 4th

November 2016. In addition, $S_X$ is related to the dimension of

covering linear spaces. Our goal is to classify Fano manifolds $X$ which

have large $S_X$.

In this talk, for an embedded Fano manifold $X$, we introduce higher

order families of lines and a new invariant $S_X$. They are line

versions of higher order minimal families of rational curves and the

invariant $N_X$ which were introduced in my previous talk on 4th

November 2016. In addition, $S_X$ is related to the dimension of

covering linear spaces. Our goal is to classify Fano manifolds $X$ which

have large $S_X$.

### 2018年04月24日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

BIRATIONAL BOUNDEDNESS OF RATIONALLY CONNECTED CALABI–YAU 3-FOLDS

(English)

**陳韋中 氏**(東大数理)BIRATIONAL BOUNDEDNESS OF RATIONALLY CONNECTED CALABI–YAU 3-FOLDS

(English)

[ 講演概要 ]

Firstly, we show that rationally connected Calabi–Yau 3- folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected 3-folds of ε-CY type form a birationally bounded family for ε > 0. Then we focus on ε-lc log Calabi–Yau pairs (X, B) such that coefficients of B are bounded from below away from zero. We show that such pairs are log bounded modulo flops. As a consequence, we show that rationally connected klt Calabi–Yau 3-folds with mld bounding away from 1 are bounded modulo flops.

Firstly, we show that rationally connected Calabi–Yau 3- folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected 3-folds of ε-CY type form a birationally bounded family for ε > 0. Then we focus on ε-lc log Calabi–Yau pairs (X, B) such that coefficients of B are bounded from below away from zero. We show that such pairs are log bounded modulo flops. As a consequence, we show that rationally connected klt Calabi–Yau 3-folds with mld bounding away from 1 are bounded modulo flops.

### 2018年04月17日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

SNC log symplectic structures on Fano products (English/Japanese)

**奥村 克彦 氏**(早稲田理工)SNC log symplectic structures on Fano products (English/Japanese)

[ 講演概要 ]

In 2014, Lima and Pereira gave a characterization of the even-dimensional projective space in terms of log symplectic Poisson structures. After that Pym gave an another more algebraic proof. In this talk, we will extend the result of Lima and Pereira to the case that the variety is a product of Fano varieties with the cyclic Picard group. This will be proved by extending Pym's proof. As a corollary, we will obtain a characterization of the projective space of all dimensions.

In 2014, Lima and Pereira gave a characterization of the even-dimensional projective space in terms of log symplectic Poisson structures. After that Pym gave an another more algebraic proof. In this talk, we will extend the result of Lima and Pereira to the case that the variety is a product of Fano varieties with the cyclic Picard group. This will be proved by extending Pym's proof. As a corollary, we will obtain a characterization of the projective space of all dimensions.

### 2018年04月09日(月)

15:30-17:00 数理科学研究科棟(駒場) 122号室

今週は月曜日にセミナーを行います。13:30-15:00と15:30-17:00の2講演あります。This week's seminar will be held on Monday and consist of two lectures: 13:30-15:00 and 15:30-17:00.

Adjoint forms on algebraic varieties (English)

今週は月曜日にセミナーを行います。13:30-15:00と15:30-17:00の2講演あります。This week's seminar will be held on Monday and consist of two lectures: 13:30-15:00 and 15:30-17:00.

**Luca Rizzi 氏**(Udine)Adjoint forms on algebraic varieties (English)

[ 講演概要 ]

The so called adjoint theory was introduced by A. Collino and G.P. Pirola in the case of smooth algebraic curves and then generalized by G.P. Pirola and F. Zucconi in the case of smooth algebraic varieties of arbitrary dimension.

The main idea of this theory is to study particular differential forms, called adjoint forms, on an algebraic variety to obtain information on the infinitesimal deformations of the variety itself.

The natural context for the application of this theory is given by Torelli-type problems, in particular infinitesimal Torelli problems.

The so called adjoint theory was introduced by A. Collino and G.P. Pirola in the case of smooth algebraic curves and then generalized by G.P. Pirola and F. Zucconi in the case of smooth algebraic varieties of arbitrary dimension.

The main idea of this theory is to study particular differential forms, called adjoint forms, on an algebraic variety to obtain information on the infinitesimal deformations of the variety itself.

The natural context for the application of this theory is given by Torelli-type problems, in particular infinitesimal Torelli problems.

### 2018年04月09日(月)

13:30-15:00 数理科学研究科棟(駒場) 122号室

今週は月曜日にセミナーを行います。13:30-15:00と15:30-17:00の2講演あります。This week's seminar will be held on Monday and consist of two lectures: 13:30-15:00 and 15:30-17:00.

Commuting nilpotents, punctual Hilbert schemes and jet bundles (ENGLISH)

今週は月曜日にセミナーを行います。13:30-15:00と15:30-17:00の2講演あります。This week's seminar will be held on Monday and consist of two lectures: 13:30-15:00 and 15:30-17:00.

**David Hyeon 氏**(ソウル大学校)Commuting nilpotents, punctual Hilbert schemes and jet bundles (ENGLISH)

[ 講演概要 ]

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers, but the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all despite its moduli theory flavor. I will explain how a 'moduli space' can be constructed via two different methods and demonstrate many interesting properties of the space:

- It is isomorphic to an open subscheme of a punctual Hilbert scheme.

- Over the field of complex numbers, it is diffeomorphic to a direct sum of twisted tangent bundles over a projective space.

- It is isomorphic to a bundle of regular jets.

- It gives examples of affine space bundles that are not vector bundles.

This is a joint work with W. Haboush (Illinois) and G. Bérczi (Zurich).

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers, but the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all despite its moduli theory flavor. I will explain how a 'moduli space' can be constructed via two different methods and demonstrate many interesting properties of the space:

- It is isomorphic to an open subscheme of a punctual Hilbert scheme.

- Over the field of complex numbers, it is diffeomorphic to a direct sum of twisted tangent bundles over a projective space.

- It is isomorphic to a bundle of regular jets.

- It gives examples of affine space bundles that are not vector bundles.

This is a joint work with W. Haboush (Illinois) and G. Bérczi (Zurich).

### 2018年01月26日(金)

16:30-18:00 数理科学研究科棟(駒場) 122号室

On classification of prime Q-Fano 3-folds with only 1/2(1,1,1)-singularities and of genus less than 2

**髙木寛通 氏**(東大数理)On classification of prime Q-Fano 3-folds with only 1/2(1,1,1)-singularities and of genus less than 2

[ 講演概要 ]

I classified prime Q-Fano threefolds with only 1/2(1,1,1)-singularities and of genus greater than 1 (2002, Nagoya Math. J.).

In this talk, I will explain how the method in that paper can be extended to the case of genus less than 2. The method is so called two ray game. By this method, I can classify the possibilities of such Q-Fano's. The classification is not yet completed since constructions of examples in certain cases are difficult. I will also explain some pretty examples in this talk.

I classified prime Q-Fano threefolds with only 1/2(1,1,1)-singularities and of genus greater than 1 (2002, Nagoya Math. J.).

In this talk, I will explain how the method in that paper can be extended to the case of genus less than 2. The method is so called two ray game. By this method, I can classify the possibilities of such Q-Fano's. The classification is not yet completed since constructions of examples in certain cases are difficult. I will also explain some pretty examples in this talk.

### 2017年12月26日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

K-stability of log Fano hyperplane arrangements (English)

**藤田 健人 氏**(RIMS)K-stability of log Fano hyperplane arrangements (English)

[ 講演概要 ]

We completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

We completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

### 2017年12月14日(木)

15:30-17:00 数理科学研究科棟(駒場) 123号室

普段と曜日・部屋が異なります

Algebraic curves and modular forms of low degree (English)

普段と曜日・部屋が異なります

**Gerard van der Geer 氏**(Universiteit van Amsterdam)Algebraic curves and modular forms of low degree (English)

[ 講演概要 ]

For genus 2 and 3 modular forms are intimately connected with the moduli of curves of genus 2 and 3. We give an explicit way to describe such modular forms for genus 2 and 3 using invariant theory and give some applications. This is based on joint work with Fabien Clery and Carel Faber.

For genus 2 and 3 modular forms are intimately connected with the moduli of curves of genus 2 and 3. We give an explicit way to describe such modular forms for genus 2 and 3 using invariant theory and give some applications. This is based on joint work with Fabien Clery and Carel Faber.

### 2017年12月14日(木)

10:30-12:00 数理科学研究科棟(駒場) 123号室

Perfectoid test ideals (English)

**Linquan Ma 氏**(University of Utah)Perfectoid test ideals (English)

[ 講演概要 ]

Inspired by the recent solution of the direct summand conjecture

of Andre and Bhatt, we introduce perfectoid multiplier/test ideals in mixed

characteristic. As an application, we obtain a uniform bound on the growth

of symbolic powers in regular local rings of mixed characteristic analogous

to results of Ein--Lazarsfeld--Smith and Hochster--Huneke in equal

characteristic. This is joint work with Karl Schwede.

Inspired by the recent solution of the direct summand conjecture

of Andre and Bhatt, we introduce perfectoid multiplier/test ideals in mixed

characteristic. As an application, we obtain a uniform bound on the growth

of symbolic powers in regular local rings of mixed characteristic analogous

to results of Ein--Lazarsfeld--Smith and Hochster--Huneke in equal

characteristic. This is joint work with Karl Schwede.

### 2017年12月05日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Ascending chain condition for F-pure thresholds on a fixed strongly F-regular germ (English or Japanese)

**佐藤 謙太 氏**(東大数理)Ascending chain condition for F-pure thresholds on a fixed strongly F-regular germ (English or Japanese)

[ 講演概要 ]

For a germ of a variety in positive characteristic and a non-zero ideal sheaf on the variety, we can define the F-pure threshold of the ideal by using Frobenius morphisms, which measures the singularities of the pair. In this talk, I will show that the set of all F-pure thresholds on a fixed strongly F-regular germ satisfies the ascending chain condition. This is a positive characteristic analogue of the "ascending chain condition for log canonical thresholds" in characteristic 0, which was recently proved by Hacon, McKernan, and Xu.

For a germ of a variety in positive characteristic and a non-zero ideal sheaf on the variety, we can define the F-pure threshold of the ideal by using Frobenius morphisms, which measures the singularities of the pair. In this talk, I will show that the set of all F-pure thresholds on a fixed strongly F-regular germ satisfies the ascending chain condition. This is a positive characteristic analogue of the "ascending chain condition for log canonical thresholds" in characteristic 0, which was recently proved by Hacon, McKernan, and Xu.

### 2017年11月28日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Kodaira vanishing theorem for Witt canonical sheaves (English)

**田中 公 氏**(東大数理)Kodaira vanishing theorem for Witt canonical sheaves (English)

[ 講演概要 ]

We establish an analogue of the Kodaira vanishing theorem in terms of de Rham-Witt complex. More specifically, given a smooth projective variety over a perfect field of positive characteristic, we prove that the higher cohomologies vanish for the tensor product of the Witt canonical sheaf and the Teichmuller lift of an ample invertible sheaf.

We establish an analogue of the Kodaira vanishing theorem in terms of de Rham-Witt complex. More specifically, given a smooth projective variety over a perfect field of positive characteristic, we prove that the higher cohomologies vanish for the tensor product of the Witt canonical sheaf and the Teichmuller lift of an ample invertible sheaf.

### 2017年11月21日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Orbifold rational connectedness (English)

**Frédéric Campana 氏**(Université de Lorraine/KIAS)Orbifold rational connectedness (English)

[ 講演概要 ]

The first step in the decomposition by canonical fibrations with fibres of `signed' canonical bundle of an arbitrary complex projective manifolds $X$ is its `rational quotient' (also called `MRC' fibration): it has rationally connected fibres and non-uniruled base. In general, the further steps (such as the Moishezon-Iitaka fibration) of this decomposition will require the consideration of 'orbifold base' of fibrations in order to deal with the multiple fibres (as seen already for elliptic surfaces). One thus needs to work in the larger category of (smooth) `orbifold pairs' $(X,D)$ to achieve this decomposition. The aim of the talk is thus to introduce the notions of Rational Connectedness and 'rational quotient' in this context, by means of suitable equivalent notions of negativity for the orbifold cotangent bundle (suitably defined. When $D$ is reduced, this is just the usual Log-version). The expected equivalence with connecting families of `orbifold rational curves' remains however presently open.

The first step in the decomposition by canonical fibrations with fibres of `signed' canonical bundle of an arbitrary complex projective manifolds $X$ is its `rational quotient' (also called `MRC' fibration): it has rationally connected fibres and non-uniruled base. In general, the further steps (such as the Moishezon-Iitaka fibration) of this decomposition will require the consideration of 'orbifold base' of fibrations in order to deal with the multiple fibres (as seen already for elliptic surfaces). One thus needs to work in the larger category of (smooth) `orbifold pairs' $(X,D)$ to achieve this decomposition. The aim of the talk is thus to introduce the notions of Rational Connectedness and 'rational quotient' in this context, by means of suitable equivalent notions of negativity for the orbifold cotangent bundle (suitably defined. When $D$ is reduced, this is just the usual Log-version). The expected equivalence with connecting families of `orbifold rational curves' remains however presently open.

### 2017年11月14日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

A characterization of the birationality of 4-canonical maps of minimal 3-folds (English)

**Meng Chen 氏**(Fudan )A characterization of the birationality of 4-canonical maps of minimal 3-folds (English)

[ 講演概要 ]

We explain the following theorem: For any minimal 3-fold X of general type with p_g>4, the 4-canonical map is non-birational if and only if X is birationally fibred by a pencil of (1,2) surfaces. The statement fails in the case of p_g=4.

We explain the following theorem: For any minimal 3-fold X of general type with p_g>4, the 4-canonical map is non-birational if and only if X is birationally fibred by a pencil of (1,2) surfaces. The statement fails in the case of p_g=4.

### 2017年11月07日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Characterizations of projective space and Seshadri constants in arbitrary characteristic

**村山 匠 氏**(ミシガン大学)Characterizations of projective space and Seshadri constants in arbitrary characteristic

[ 講演概要 ]

Mori and Mukai conjectured that projective space should be the only n-dimensional Fano variety whose anti-canonical bundle has degree at least n + 1 along every curve. While this conjecture has been proved in characteristic zero, it remains open in positive characteristic. We will present some progress in this direction by giving another characterization of projective space using Seshadri constants and the Frobenius morphism. The key ingredient is a positive-characteristic analogue of Demailly’s criterion for separation of higher-order jets by adjoint bundles, whose proof gives new results for adjoint bundles even in characteristic zero.

Mori and Mukai conjectured that projective space should be the only n-dimensional Fano variety whose anti-canonical bundle has degree at least n + 1 along every curve. While this conjecture has been proved in characteristic zero, it remains open in positive characteristic. We will present some progress in this direction by giving another characterization of projective space using Seshadri constants and the Frobenius morphism. The key ingredient is a positive-characteristic analogue of Demailly’s criterion for separation of higher-order jets by adjoint bundles, whose proof gives new results for adjoint bundles even in characteristic zero.

### 2017年10月31日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

ACC for log canonical threshold polytopes (English)

**Zhan Li 氏**(Beijing)ACC for log canonical threshold polytopes (English)

[ 講演概要 ]

We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition. This is a joint work with Jingjun Han and Lu Qi.

We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition. This is a joint work with Jingjun Han and Lu Qi.

### 2017年10月30日(月)

10:30-12:00 数理科学研究科棟(駒場) 123号室

Towards birational boundedness of elliptic Calabi-Yau varieties (English)

**Roberto Svaldi 氏**(Cambridge)Towards birational boundedness of elliptic Calabi-Yau varieties (English)

[ 講演概要 ]

I will discuss new results towards the birational boundedness of

low-dimensional elliptic Calabi-Yau varieties, joint work with Gabriele

Di Certo.

Recent work in the minimal model program suggests that pairs with trivial log canonical

class should satisfy some boundedness properties.

I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are

indeed log birationally bounded. This implies birational boundedness of elliptically fibered

Calabi-Yau manifolds with a section, in dimension up to 5.

If time allows, I will also try to discuss a first approach towards boundedness of rationally

connected CY varieties in low dimension.

I will discuss new results towards the birational boundedness of

low-dimensional elliptic Calabi-Yau varieties, joint work with Gabriele

Di Certo.

Recent work in the minimal model program suggests that pairs with trivial log canonical

class should satisfy some boundedness properties.

I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are

indeed log birationally bounded. This implies birational boundedness of elliptically fibered

Calabi-Yau manifolds with a section, in dimension up to 5.

If time allows, I will also try to discuss a first approach towards boundedness of rationally

connected CY varieties in low dimension.

### 2017年10月17日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Intersection of currents, dimension excess and complex dynamics (English)

**Tien Cuong Dinh 氏**(Singapore)Intersection of currents, dimension excess and complex dynamics (English)

[ 講演概要 ]

I will discuss dynamical properties of Henon maps in higher dimension, in particular, the equidistribution property of periodic points. Positive closed currents can be seen as an analytic counterpart of effective algebraic cycles. I will explain how a non-generic intersection theory for these currents, possibly with dimension excess, comes into the picture. Other applications of the intersection theory will be also discussed. This is a joint work with Nessim Sibony.

I will discuss dynamical properties of Henon maps in higher dimension, in particular, the equidistribution property of periodic points. Positive closed currents can be seen as an analytic counterpart of effective algebraic cycles. I will explain how a non-generic intersection theory for these currents, possibly with dimension excess, comes into the picture. Other applications of the intersection theory will be also discussed. This is a joint work with Nessim Sibony.

### 2017年10月10日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Classification of Mukai pairs with corank 3 (English or Japanese)

**金光 秋博 氏**(東大数理)Classification of Mukai pairs with corank 3 (English or Japanese)

[ 講演概要 ]

A Mukai pair $(X,E)$ is a pair of a Fano manifold $X$ and an ample vector bundle $E$ of rank $r$ on $X$ such that $c_1(X)=c_1(E)$. Study of such pairs was proposed by Mukai. It is known that, for a Mukai pair $(X,E)$, the rank $r$ of the bundle $E$ is at most $\dim X +1$, and Mukai conjectured the explicit

classification with $r \geq \dim X$. The above conjecture was solved independently by Fujita, Peternell and Ye-Zhang. Also the classification of Mukai pairs with $r= \dim X -1$ was given by Peternell-Szurek-Wi\'sniewski. In this talk I will give the classification of Mukai pairs with $r= \dim X -2$ and $\dim X \geq 5$.

A Mukai pair $(X,E)$ is a pair of a Fano manifold $X$ and an ample vector bundle $E$ of rank $r$ on $X$ such that $c_1(X)=c_1(E)$. Study of such pairs was proposed by Mukai. It is known that, for a Mukai pair $(X,E)$, the rank $r$ of the bundle $E$ is at most $\dim X +1$, and Mukai conjectured the explicit

classification with $r \geq \dim X$. The above conjecture was solved independently by Fujita, Peternell and Ye-Zhang. Also the classification of Mukai pairs with $r= \dim X -1$ was given by Peternell-Szurek-Wi\'sniewski. In this talk I will give the classification of Mukai pairs with $r= \dim X -2$ and $\dim X \geq 5$.

### 2017年07月18日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On a generalization of Frobenius-splitting and a lifting problem of Calabi-Yau varieties (JAPANESE)

**呼子 笛太郎 氏**(東北大理)On a generalization of Frobenius-splitting and a lifting problem of Calabi-Yau varieties (JAPANESE)

[ 講演概要 ]

In this talk, we introduce a notion of Frobenius-splitting height which quantifies Frobenius-splitting varieties and show that a Calabi-Yau variety of finite height over an algebraically closed field of positive characteristic admits a flat lifting to the ring of Witt vectors of length two.

In this talk, we introduce a notion of Frobenius-splitting height which quantifies Frobenius-splitting varieties and show that a Calabi-Yau variety of finite height over an algebraically closed field of positive characteristic admits a flat lifting to the ring of Witt vectors of length two.

### 2017年07月11日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Arithmetic and dynamical degrees of self-maps of algebraic varieties (English or Japanese)

**松澤 陽介 氏**(東大数理)Arithmetic and dynamical degrees of self-maps of algebraic varieties (English or Japanese)

[ 講演概要 ]

The first dynamical degree is an important birational invariant which measures the geometric complexity of dominant rational self-maps of algebraic varieties. On the other hand, when the variety is defined over a number field, one can associate to an orbit an invariant using Weil height function, called arithmetic degree, which measures the arithmetic complexity of the orbit. It is conjectured that the arithmetic degree of a Zariski dense orbit is equal to the first dynamical degree (Kawaguchi-Silverman). I will explain several results related to this conjecture. I will also explain applications to proofs of purely geometric statements.

The first dynamical degree is an important birational invariant which measures the geometric complexity of dominant rational self-maps of algebraic varieties. On the other hand, when the variety is defined over a number field, one can associate to an orbit an invariant using Weil height function, called arithmetic degree, which measures the arithmetic complexity of the orbit. It is conjectured that the arithmetic degree of a Zariski dense orbit is equal to the first dynamical degree (Kawaguchi-Silverman). I will explain several results related to this conjecture. I will also explain applications to proofs of purely geometric statements.

### 2017年07月04日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

The space of rational curves and Manin's conjecture (English)

**谷本 祥 氏**(University of Copenhagen)The space of rational curves and Manin's conjecture (English)

[ 講演概要 ]

Manin's conjecture predicts the asymptotic formula for the counting function of rational points on a Fano variety after removing the exceptional thin set. There are many developments on birational geometry of exceptional sets using MMP, due to Lehmann, myself, Tschinkel, Hacon, and Jiang. Recently we found that the study of exceptional sets has applications to questions regarding the space of rational curves, i.e., its dimension and the number of components. I would like to explain these applications. This is joint work with Brian Lehmann.

Manin's conjecture predicts the asymptotic formula for the counting function of rational points on a Fano variety after removing the exceptional thin set. There are many developments on birational geometry of exceptional sets using MMP, due to Lehmann, myself, Tschinkel, Hacon, and Jiang. Recently we found that the study of exceptional sets has applications to questions regarding the space of rational curves, i.e., its dimension and the number of components. I would like to explain these applications. This is joint work with Brian Lehmann.

### 2017年06月27日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Cylinders in del Pezzo fibrations (English )

**岸本 崇 氏**(埼玉大学)Cylinders in del Pezzo fibrations (English )

[ 講演概要 ]

The cylinder is, by definition, an algebraic variety of the form Z × A1 . Certainly it is geometrically a very simple object, but it plays often an important role to connect unipotent group actions on special kinds of affine algebraic varieties to projective geometry. From the point of view of birational geometry, it is essential to look into cylinders found on Mori fiber spaces. In this talk, we shall focus mainly on Mori fiber spaces of relative dimension two or three. One of main results asserts that a del Pezzo fibration π : V → W contains a cylinder respecting the structure of π (so-called a vertical cylinder) if and only if the degree deg π of π is greater than or equal to 5 and π admits a rational section. Especially, in case of dim V = 3, the existence of a vertical cylinder is equivalent to saying deg π ≧ 5 in consideration of Tsen’s theorem, nevertheless, it is worthwhile to note that the affine 3-space A3C is embedded into certains del Pezzo fibrations π : V → P1C of deg π ≦ 4 in a twisted way. This is a joint work with Adrien Dubouloz (Universit ́e de Bourgogne).

The cylinder is, by definition, an algebraic variety of the form Z × A1 . Certainly it is geometrically a very simple object, but it plays often an important role to connect unipotent group actions on special kinds of affine algebraic varieties to projective geometry. From the point of view of birational geometry, it is essential to look into cylinders found on Mori fiber spaces. In this talk, we shall focus mainly on Mori fiber spaces of relative dimension two or three. One of main results asserts that a del Pezzo fibration π : V → W contains a cylinder respecting the structure of π (so-called a vertical cylinder) if and only if the degree deg π of π is greater than or equal to 5 and π admits a rational section. Especially, in case of dim V = 3, the existence of a vertical cylinder is equivalent to saying deg π ≧ 5 in consideration of Tsen’s theorem, nevertheless, it is worthwhile to note that the affine 3-space A3C is embedded into certains del Pezzo fibrations π : V → P1C of deg π ≦ 4 in a twisted way. This is a joint work with Adrien Dubouloz (Universit ́e de Bourgogne).

### 2017年06月12日(月)

17:00-18:30 数理科学研究科棟(駒場) 056号室

普段と曜日・部屋が異なります

Rational and irrational singular quartic threefolds (English)

普段と曜日・部屋が異なります

**Ivan Cheltsov 氏**(The University of Edinburgh)Rational and irrational singular quartic threefolds (English)

[ 講演概要 ]

Burkhardt and Igusa quartics admit a faithful action of the symmetric group of degree 6.

There are other quartic threefolds with this property. All of them are singular.

Beauville proved that all but four of them are irrational. Burkhardt and Igusa quartics are known to be rational.

Two constructions of Todd imply the rationality of the remaining two quartic threefolds.

In this talk, I will give an alternative proof of both these (irrationality and rationality) results.

This proof is based on explicit small resolutions of the so-called Coble fourfold.

This fourfold is the double cover of the four-dimensional projective space branched over Igusa quartic.

This is a joint work with Sasha Kuznetsov and Costya Shramov.

Burkhardt and Igusa quartics admit a faithful action of the symmetric group of degree 6.

There are other quartic threefolds with this property. All of them are singular.

Beauville proved that all but four of them are irrational. Burkhardt and Igusa quartics are known to be rational.

Two constructions of Todd imply the rationality of the remaining two quartic threefolds.

In this talk, I will give an alternative proof of both these (irrationality and rationality) results.

This proof is based on explicit small resolutions of the so-called Coble fourfold.

This fourfold is the double cover of the four-dimensional projective space branched over Igusa quartic.

This is a joint work with Sasha Kuznetsov and Costya Shramov.