## Algebraic Geometry Seminar

Seminar information archive ～09/27｜Next seminar｜Future seminars 09/28～

Date, time & place | Tuesday 10:30 - 11:30 or 12:00 ハイブリッド開催/002Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.) |
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**Seminar information archive**

### 2018/01/26

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Hiromichi Takagi**(The University of Tokyo)On classification of prime Q-Fano 3-folds with only 1/2(1,1,1)-singularities and of genus less than 2

[ Abstract ]

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.

### 2017/12/26

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

K-stability of log Fano hyperplane arrangements (English)

**Kento Fujita**(RIMS)K-stability of log Fano hyperplane arrangements (English)

[ Abstract ]

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.

### 2017/12/14

15:30-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Algebraic curves and modular forms of low degree (English)

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

[ Abstract ]

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.

### 2017/12/14

10:30-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Perfectoid test ideals (English)

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

[ Abstract ]

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.

### 2017/12/05

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Kenta Sato**(The University of Tokyo)Ascending chain condition for F-pure thresholds on a fixed strongly F-regular germ (English or Japanese)

[ Abstract ]

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.

### 2017/11/28

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Kodaira vanishing theorem for Witt canonical sheaves (English)

**Hiromu Tanaka**(Tokyo)Kodaira vanishing theorem for Witt canonical sheaves (English)

[ Abstract ]

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.

### 2017/11/21

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Orbifold rational connectedness (English)

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

[ Abstract ]

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.

### 2017/11/14

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2017/11/07

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Characterizations of projective space and Seshadri constants in arbitrary characteristic

**Takumi Murayama**(University of Michigan)Characterizations of projective space and Seshadri constants in arbitrary characteristic

[ Abstract ]

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.

### 2017/10/31

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

ACC for log canonical threshold polytopes (English)

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

[ Abstract ]

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.

### 2017/10/30

10:30-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2017/10/17

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2017/10/10

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Akihiro Kanemitsu**(The University of Tokyo)Classification of Mukai pairs with corank 3 (English or Japanese)

[ Abstract ]

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 $\dim X \geq 5$.

### 2017/07/18

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Fuetaro Yobuko**(Tohoku University)On a generalization of Frobenius-splitting and a lifting problem of Calabi-Yau varieties (JAPANESE)

[ Abstract ]

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.

### 2017/07/11

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Yohsuke Matsuzawa**(The University of Tokyo)Arithmetic and dynamical degrees of self-maps of algebraic varieties (English or Japanese)

[ Abstract ]

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.

### 2017/07/04

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Sho Tanimoto**(University of Copenhagen)The space of rational curves and Manin’s conjecture (English)

[ Abstract ]

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.

### 2017/06/27

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Cylinders in del Pezzo fibrations (English )

**Takashi Kishimoto**(Saitama University)Cylinders in del Pezzo fibrations (English )

[ Abstract ]

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).

### 2017/06/12

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Rational and irrational singular quartic threefolds (English)

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

[ Abstract ]

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.

### 2017/06/06

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Fano varieties: K-stability and boundedness (English)

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

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

[ Abstract ]

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.

[ Reference 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/

### 2017/05/30

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Masaru Nagaoka**(The University of Tokyo)Contractible affine threefolds in smooth Fano threefolds (English or Japanese)

[ Abstract ]

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.

### 2017/05/23

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Naoki Koseki**(The University of Tokyo)Perverse coherent sheaves on blow-ups at codimension two loci (English)

[ Abstract ]

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.

### 2017/05/16

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On separable higher Gauss maps (English)

**Katsuhisa Furukawa**(The University of Tokyo)On separable higher Gauss maps (English)

[ Abstract ]

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.

### 2017/05/09

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Kohsuke Shibata**(The University of Tokyo)Upper bound of the multiplicity of locally complete intersection singularities (English)

[ Abstract ]

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.

### 2017/04/25

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Taku Suzuki**(Waseda University)On the Picard number of Fano 6-folds with a non-small contraction (English)

[ Abstract ]

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.

### 2017/04/18

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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

**Takeru Fukuoka**(The University of Tokyo)On the existence of almost Fano threefolds with del Pezzo fibrations (English)

[ Abstract ]

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.