## 代数幾何学セミナー

過去の記録 ～02/23｜次回の予定｜今後の予定 02/24～

開催情報 | 月曜日 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・高木 俊輔 |

**過去の記録**

### 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.

### 2017年06月06日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Fano varieties: K-stability and boundedness (English)

https://sites.google.com/site/chenjiangmath/

**Chen Jiang 氏**(IPMU)Fano varieties: K-stability and boundedness (English)

[ 講演概要 ]

There are two interesting problems for Fano varieties, K-stability and boundedness.

Significant progress has been made for both problems recently.

In this talk, I will show the boundedness of K-semistable Fano varieties with anti-canonical degree bounded from below, by using methods from birational geometry.

[ 講演参考URL ]There are two interesting problems for Fano varieties, K-stability and boundedness.

Significant progress has been made for both problems recently.

In this talk, I will show the boundedness of K-semistable Fano varieties with anti-canonical degree bounded from below, by using methods from birational geometry.

https://sites.google.com/site/chenjiangmath/

### 2017年05月30日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Contractible affine threefolds in smooth Fano threefolds (English or Japanese)

**長岡 大 氏**(東大数理)Contractible affine threefolds in smooth Fano threefolds (English or Japanese)

[ 講演概要 ]

By the contribution of M. Furushima, N. Nakayama, Th. Peternell and M.

Schneider, it is completed to classify all projective compactifications

of the affine $3$-space $\mathbb{A}^3$ with Picard number one.

As a similar question, T. Kishimoto raised the problem to classify all

triplets $(V, U, D_1 \cup D_2)$ which consist of smooth Fano threefolds

$V$ of Picard number two, contractible affine threefolds $U$ as open

subsets of $V$, and the complements $D_1 \cup D_2 =V \setminus U$.

He also solved this problem when the log canonical divisors $K_V+D_1+D_2

$ are not nef.

In this talk, I will discuss the triplets $(V, U, D_1 \cup D_2)$ whose

log canonical divisors are linearly equivalent to zero.

I will also explain how to determine all Fano threefolds $V$ which

appear in such triplets.

By the contribution of M. Furushima, N. Nakayama, Th. Peternell and M.

Schneider, it is completed to classify all projective compactifications

of the affine $3$-space $\mathbb{A}^3$ with Picard number one.

As a similar question, T. Kishimoto raised the problem to classify all

triplets $(V, U, D_1 \cup D_2)$ which consist of smooth Fano threefolds

$V$ of Picard number two, contractible affine threefolds $U$ as open

subsets of $V$, and the complements $D_1 \cup D_2 =V \setminus U$.

He also solved this problem when the log canonical divisors $K_V+D_1+D_2

$ are not nef.

In this talk, I will discuss the triplets $(V, U, D_1 \cup D_2)$ whose

log canonical divisors are linearly equivalent to zero.

I will also explain how to determine all Fano threefolds $V$ which

appear in such triplets.

### 2017年05月23日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Perverse coherent sheaves on blow-ups at codimension two loci (English)

**小関 直紀 氏**(東大数理)Perverse coherent sheaves on blow-ups at codimension two loci (English)

[ 講演概要 ]

I would like to talk about my recent work in progress.

Let us consider the blow-up X of Y along a subvariety C.

Then the following natural question arises:

What is the relation between moduli space of sheaves on Y

and that of X?

H.Nakajima and K.Yoshioka answered the above question

in the case when Y is a surface and C is a point. They

showed that the moduli spaces are connected by a sequence

of flip-like diagrams. The key ingredient of the proof is

to use perverse coherent sheaves in the sense of T.Bridgeland

and M.Van den Bergh.

In this talk, I will explain how to generalize their theorem

to the case when Y is a smooth projective variety of arbitrary

dimension and C is its codimension two subvariety.

I would like to talk about my recent work in progress.

Let us consider the blow-up X of Y along a subvariety C.

Then the following natural question arises:

What is the relation between moduli space of sheaves on Y

and that of X?

H.Nakajima and K.Yoshioka answered the above question

in the case when Y is a surface and C is a point. They

showed that the moduli spaces are connected by a sequence

of flip-like diagrams. The key ingredient of the proof is

to use perverse coherent sheaves in the sense of T.Bridgeland

and M.Van den Bergh.

In this talk, I will explain how to generalize their theorem

to the case when Y is a smooth projective variety of arbitrary

dimension and C is its codimension two subvariety.

### 2017年05月16日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On separable higher Gauss maps (English)

**古川 勝久 氏**(東大数理)On separable higher Gauss maps (English)

[ 講演概要 ]

We study the $m$-th Gauss map in the sense of F. L. Zak of a projective variety $X ¥subset P^N$ over an algebraically closed field in any characteristic, where $m$ is an integer with $n:= ¥dim(X) ¥leq m < N$. It is known that the contact locus on $X$ of a general tangent $m$-plane can be non-linear in positive characteristic, if the $m$-th Gauss map is inseparable.

In this talk, I will explain that for any $m$, the locus is a linear variety if the $m$-th Gauss map is separable. I will also explain that for smooth $X$ with $n < N-2$, the $(n+1)$-th Gauss

map is birational if it is separable, unless $X$ is the Segre embedding $P^1 ¥times P^n ¥subset P^{2n-1}$. This is related to L. Ein's classification of varieties with small dual varieties in characteristic zero.

This talk is based on a joint work with Atsushi Ito.

We study the $m$-th Gauss map in the sense of F. L. Zak of a projective variety $X ¥subset P^N$ over an algebraically closed field in any characteristic, where $m$ is an integer with $n:= ¥dim(X) ¥leq m < N$. It is known that the contact locus on $X$ of a general tangent $m$-plane can be non-linear in positive characteristic, if the $m$-th Gauss map is inseparable.

In this talk, I will explain that for any $m$, the locus is a linear variety if the $m$-th Gauss map is separable. I will also explain that for smooth $X$ with $n < N-2$, the $(n+1)$-th Gauss

map is birational if it is separable, unless $X$ is the Segre embedding $P^1 ¥times P^n ¥subset P^{2n-1}$. This is related to L. Ein's classification of varieties with small dual varieties in characteristic zero.

This talk is based on a joint work with Atsushi Ito.

### 2017年05月09日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

Upper bound of the multiplicity of locally complete intersection singularities (English)

**柴田 康介 氏**(東大数理)Upper bound of the multiplicity of locally complete intersection singularities (English)

[ 講演概要 ]

The multiplicity of a point on a variety is a fundamental invariant to estimate how the singularity is bad. It is introduced in a purely algebraic context. On the other hand, we can also attach to the singularity the log canonical threshold and the minimal log discrepancy, which are introduced in a birational theoretic context. In this talk, we show bounds of the multiplicity by functions of these birational invariants for a singularity of locally a complete intersection. As an application, we obtain the affirmative answer to Watanabe’s conjecture on the multiplicity of canonical singularity of locally a complete intersection up to dimension 32.

The multiplicity of a point on a variety is a fundamental invariant to estimate how the singularity is bad. It is introduced in a purely algebraic context. On the other hand, we can also attach to the singularity the log canonical threshold and the minimal log discrepancy, which are introduced in a birational theoretic context. In this talk, we show bounds of the multiplicity by functions of these birational invariants for a singularity of locally a complete intersection. As an application, we obtain the affirmative answer to Watanabe’s conjecture on the multiplicity of canonical singularity of locally a complete intersection up to dimension 32.

### 2017年04月25日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On the Picard number of Fano 6-folds with a non-small contraction (English)

**鈴木 拓 氏**(早稲田大学)On the Picard number of Fano 6-folds with a non-small contraction (English)

[ 講演概要 ]

A generalization of S. Mukai's conjecture says that $\rho(i-1) \leq n$ holds for any Fano $n$-fold with Picard number $\rho$ and pseudo-index $i$, with equality if and only if it is isomorphic to $(\mathbb{P}^{i-1})^{\rho}$. In this talk, we consider this conjecture for $n=6$, which is an open problem, and give a proof of some special cases.

A generalization of S. Mukai's conjecture says that $\rho(i-1) \leq n$ holds for any Fano $n$-fold with Picard number $\rho$ and pseudo-index $i$, with equality if and only if it is isomorphic to $(\mathbb{P}^{i-1})^{\rho}$. In this talk, we consider this conjecture for $n=6$, which is an open problem, and give a proof of some special cases.

### 2017年04月18日(火)

15:30-17:00 数理科学研究科棟(駒場) 122号室

On the existence of almost Fano threefolds with del Pezzo fibrations (English)

**福岡 尊 氏**(東大数理)On the existence of almost Fano threefolds with del Pezzo fibrations (English)

[ 講演概要 ]

We say that a smooth projective 3-fold is almost Fano if its anti-canonical divisor is nef and big but not ample. By Jahnke-Peternell-Radloff and Takeuchi, the numerical classification of such 3-folds was given. Among the classification results, there exists precisely 10 cases such that it was yet to be known whether these have an example or not. The main result of this talk shows the existence of examples of each of 10 cases. In 9 cases of the 10 cases, the degree of del Pezzo fibrations are 6. We will discuss one of the reason of difficulty constructing del Pezzo fibrations of degree 6. After that, we will show that every almost Fano del Pezzo fibration of degree 6 with specific anti-canonical volume can be embedded into some higher dimensional del Pezzo fibration as a relative linear section.

We say that a smooth projective 3-fold is almost Fano if its anti-canonical divisor is nef and big but not ample. By Jahnke-Peternell-Radloff and Takeuchi, the numerical classification of such 3-folds was given. Among the classification results, there exists precisely 10 cases such that it was yet to be known whether these have an example or not. The main result of this talk shows the existence of examples of each of 10 cases. In 9 cases of the 10 cases, the degree of del Pezzo fibrations are 6. We will discuss one of the reason of difficulty constructing del Pezzo fibrations of degree 6. After that, we will show that every almost Fano del Pezzo fibration of degree 6 with specific anti-canonical volume can be embedded into some higher dimensional del Pezzo fibration as a relative linear section.