## Algebraic Geometry Seminar

Seminar information archive ～02/07｜Next seminar｜Future seminars 02/08～

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**

### 2010/11/16

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

Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)

**Viacheslav Nikulin**(Univ Liverpool and Steklov Moscow)Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)

[ Abstract ]

In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here, we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index. See some details in arXiv:0810.2945.

In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here, we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index. See some details in arXiv:0810.2945.

### 2010/11/15

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Generators of tropical modules (JAPANESE)

**Shuhei Yoshitomi**(Univ. of Tokyo)Generators of tropical modules (JAPANESE)

### 2010/11/01

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

How to estimate Seshadri constants (JAPANESE)

**Atsushi Ito**(Univ. of Tokyo)How to estimate Seshadri constants (JAPANESE)

[ Abstract ]

Seshadri constant is an invariant which measures the positivities of ample line bundles. This relates with adjoint bundles, Nagata conjecture, slope stabilities, Gromov width (an invariant of symplectic manifolds) and so on. But it is very diffiult to compute or estimate Seshadri constants in general, especially in higher dimension.

In this talk, we first study Seshadri constants of toric varieties, and next consider about non-toric cases using toric degenerations. For example, good estimations are obtained for complete intersections in projective spaces.

Seshadri constant is an invariant which measures the positivities of ample line bundles. This relates with adjoint bundles, Nagata conjecture, slope stabilities, Gromov width (an invariant of symplectic manifolds) and so on. But it is very diffiult to compute or estimate Seshadri constants in general, especially in higher dimension.

In this talk, we first study Seshadri constants of toric varieties, and next consider about non-toric cases using toric degenerations. For example, good estimations are obtained for complete intersections in projective spaces.

### 2010/10/18

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Galois extensions and maps on local cohomology (JAPANESE)

**Akiyoshi Sannai**(Univ. of Tokyo)Galois extensions and maps on local cohomology (JAPANESE)

### 2010/09/06

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Non-reduced components of the Noether-Lefschetz locus (ENGLISH)

**Prof. Remke Kloosterman**(Humboldt University, Berlin)Non-reduced components of the Noether-Lefschetz locus (ENGLISH)

[ Abstract ]

Let $M_d$ be the moduli space of complex smooth degree $d$ surfaces in $\\mathbb{P}3$. Let $NL_d \\subset M_d$ be the subset corresponding to surfaces with Picard number at least 2. It is known that $NL_r$ is Zariski-constructable, and each irreducible component of $NL_r$ has a natural scheme structure. In this talk we describe the largest non-reduced components of $NL_r$. This extends work of Maclean and Otwinowska.

This is joint work with my PhD student Ananyo Dan.

Let $M_d$ be the moduli space of complex smooth degree $d$ surfaces in $\\mathbb{P}3$. Let $NL_d \\subset M_d$ be the subset corresponding to surfaces with Picard number at least 2. It is known that $NL_r$ is Zariski-constructable, and each irreducible component of $NL_r$ has a natural scheme structure. In this talk we describe the largest non-reduced components of $NL_r$. This extends work of Maclean and Otwinowska.

This is joint work with my PhD student Ananyo Dan.

### 2010/07/29

14:30-16:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Homological Mirror Symmetry for 2-dimensional toric Fano stacks (JAPANESE)

**Masahiro Futaki**(The University of Tokyo)Homological Mirror Symmetry for 2-dimensional toric Fano stacks (JAPANESE)

[ Abstract ]

Homological Mirror Symmetry (HMS for short) is a conjectural

duality between complex and symplectic geometry, originally proposed

for mirror pairs of Calabi-Yau manifolds and later extended to

Fano/Landau-Ginzburg mirrors (both due to Kontsevich, 1994 and 1998).

We explain how HMS is established in the case of 2-dimensional smooth

toric Fano stack X as an equivalence between the derived category of X

and the derived directed Fukaya category of its mirror Lefschetz

fibration W. This is related to Kontsevich-Soibelman's construction of

3d CY category from the quiver with potential.

We also obtain a local mirror extension following Seidel's suspension

theorem, that is, the local HMS for the canonical bundle K_X and the

double suspension W+uv. This talk is joint with Kazushi Ueda (Osaka

U.).

Homological Mirror Symmetry (HMS for short) is a conjectural

duality between complex and symplectic geometry, originally proposed

for mirror pairs of Calabi-Yau manifolds and later extended to

Fano/Landau-Ginzburg mirrors (both due to Kontsevich, 1994 and 1998).

We explain how HMS is established in the case of 2-dimensional smooth

toric Fano stack X as an equivalence between the derived category of X

and the derived directed Fukaya category of its mirror Lefschetz

fibration W. This is related to Kontsevich-Soibelman's construction of

3d CY category from the quiver with potential.

We also obtain a local mirror extension following Seidel's suspension

theorem, that is, the local HMS for the canonical bundle K_X and the

double suspension W+uv. This talk is joint with Kazushi Ueda (Osaka

U.).

### 2010/07/12

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Flips of moduli of stable torsion free sheaves with $c_1=1$ on

$\\mathbb{P}^2$ (JAPANESE)

**Ryo Ohkawa**(Tokyo Institute of Technology)Flips of moduli of stable torsion free sheaves with $c_1=1$ on

$\\mathbb{P}^2$ (JAPANESE)

[ Abstract ]

We study flips of moduli schemes of stable torsion free sheaves

on the projective plane via wall-crossing phenomena of Bridgeland stability.

They are described as stratified Grassmann bundles by variation of

stability of modules over certain finite dimensional algebra.

We study flips of moduli schemes of stable torsion free sheaves

on the projective plane via wall-crossing phenomena of Bridgeland stability.

They are described as stratified Grassmann bundles by variation of

stability of modules over certain finite dimensional algebra.

### 2010/07/05

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Rational curves on hypersurfaces (JAPANESE)

**Katsuhisa Furukawa**(Waseda University)Rational curves on hypersurfaces (JAPANESE)

[ Abstract ]

Our purpose is to study the family of smooth rational curves of degree $e$ lying on a hypersurface of degree $d$ in $\\mathbb{P}^n$, and to investigate properties of this family (e.g., dimension, smoothness, connectedness).

Our starting point is the research about the family of lines (i.e., $e = 1$), which was studied by W. Barth and A. Van de Ven over $\\mathbb{C}$, and by J. Koll\\'{a}r over an algebraically closed field of arbitrary characteristic.

For the degree $e > 1$, the family of rational curves was studied by J. Harris, M. Roth, and J. Starr over $\\mathbb{C}$ in the case of $d < (n+1)/2$.

In this talk, we study the family of rational curves in arbitrary characteristic under the assumption $e = 2,3$ and $d > 1$, or $e > 3$ and $d > 2e-4$.

Our purpose is to study the family of smooth rational curves of degree $e$ lying on a hypersurface of degree $d$ in $\\mathbb{P}^n$, and to investigate properties of this family (e.g., dimension, smoothness, connectedness).

Our starting point is the research about the family of lines (i.e., $e = 1$), which was studied by W. Barth and A. Van de Ven over $\\mathbb{C}$, and by J. Koll\\'{a}r over an algebraically closed field of arbitrary characteristic.

For the degree $e > 1$, the family of rational curves was studied by J. Harris, M. Roth, and J. Starr over $\\mathbb{C}$ in the case of $d < (n+1)/2$.

In this talk, we study the family of rational curves in arbitrary characteristic under the assumption $e = 2,3$ and $d > 1$, or $e > 3$ and $d > 2e-4$.

### 2010/06/21

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Pseudo-index and minimal length of extremal rays for Fano manifolds (JAPANESE)

**Toru Tsukioka**(Osaka Prefecture University)Pseudo-index and minimal length of extremal rays for Fano manifolds (JAPANESE)

[ Abstract ]

The minimum of intersection numbers of the anticanonical

divisor with rational curves on a Fano manifold is called pseudo-index.

In view of the fact that the geometry of Fano manifolds is governed by

its extremal rays, it is important to consider the extremal rational

curves. In this talk, we show that for Fano 4-folds having birational

contractions, the minimal length of extremal rays coincides with the

pseudo-index.

The minimum of intersection numbers of the anticanonical

divisor with rational curves on a Fano manifold is called pseudo-index.

In view of the fact that the geometry of Fano manifolds is governed by

its extremal rays, it is important to consider the extremal rational

curves. In this talk, we show that for Fano 4-folds having birational

contractions, the minimal length of extremal rays coincides with the

pseudo-index.

### 2010/06/14

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Slope of smooth rational curves in an anticanonically polarized Fano manifold (ENGLISH)

**Yongnam Lee**(Sogang University)Slope of smooth rational curves in an anticanonically polarized Fano manifold (ENGLISH)

[ Abstract ]

Ross and Thomas introduce the concept of slope stability to study K-stability, which has conjectural relation with the existence of constant scalar curvature metric. Since K-stability implies slope stability, slope stability gives an algebraic obstruction to theexistence of constant scalar curvature. This talk presents a systematic study of slope stability of anticanonically polarized Fano manifolds with respect to smooth rational curves. Especially, we prove that an anticanonically polarized Fano maniold is slope semistable with respect to any free smooth rational curves, and that an anticanonically polarized Fano threefold X with Picard number 1 is slope stable with respect to any smooth rational curves unless X is the project space. It is a joint work with Jun-Muk Hwang and Hosung Kim.

Ross and Thomas introduce the concept of slope stability to study K-stability, which has conjectural relation with the existence of constant scalar curvature metric. Since K-stability implies slope stability, slope stability gives an algebraic obstruction to theexistence of constant scalar curvature. This talk presents a systematic study of slope stability of anticanonically polarized Fano manifolds with respect to smooth rational curves. Especially, we prove that an anticanonically polarized Fano maniold is slope semistable with respect to any free smooth rational curves, and that an anticanonically polarized Fano threefold X with Picard number 1 is slope stable with respect to any smooth rational curves unless X is the project space. It is a joint work with Jun-Muk Hwang and Hosung Kim.

### 2010/06/07

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Genus 2 curve configurations on Fano surfaces (ENGLISH)

**Xavier Roulleau**(The University of Tokyo)Genus 2 curve configurations on Fano surfaces (ENGLISH)

### 2010/05/31

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

On Pfaffian Calabi-Yau Varieties and Mirror Symmetry (JAPANESE)

**Atsushi Kanazawa**(The University of Tokyo)On Pfaffian Calabi-Yau Varieties and Mirror Symmetry (JAPANESE)

[ Abstract ]

We construct new smooth CY 3-folds with 1-dimensional Kaehler moduli and

determine their fundamental topological invariants. The existence of CY

3-folds with the computed invariants was previously conjectured. We then

report mirror symmetry for these non-complete intersection CY 3-folds.

We explicitly build their mirror partners, some of which have 2 LCSLs,

and carry out instanton computations for g=0,1.

We construct new smooth CY 3-folds with 1-dimensional Kaehler moduli and

determine their fundamental topological invariants. The existence of CY

3-folds with the computed invariants was previously conjectured. We then

report mirror symmetry for these non-complete intersection CY 3-folds.

We explicitly build their mirror partners, some of which have 2 LCSLs,

and carry out instanton computations for g=0,1.

### 2010/05/24

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

A counterexample of the birational Torelli problem via Fourier--Mukai transforms (JAPANESE)

**Hokuto Uehara**(Tokyo Metropolitan University)A counterexample of the birational Torelli problem via Fourier--Mukai transforms (JAPANESE)

[ Abstract ]

We study the Fourier--Mukai numbers of rational elliptic surfaces. As

its application, we give an example of a pair of minimal 3-folds $X$

with Kodaira dimensions 1, $h^1(O_X)=h^2(O_X)=0$ such that they are

mutually derived equivalent, deformation equivalent, but not

birationally equivalent. It also supplies a counterexample of the

birational Torelli problem.

We study the Fourier--Mukai numbers of rational elliptic surfaces. As

its application, we give an example of a pair of minimal 3-folds $X$

with Kodaira dimensions 1, $h^1(O_X)=h^2(O_X)=0$ such that they are

mutually derived equivalent, deformation equivalent, but not

birationally equivalent. It also supplies a counterexample of the

birational Torelli problem.

### 2010/05/17

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

On the GIT stability of Polarized Varieties (JAPANESE)

**Yuji Odaka**(Research Institute for Mathematical Sciences)On the GIT stability of Polarized Varieties (JAPANESE)

[ Abstract ]

Background:

Original GIT-stability notion for polarized variety is

"asymptotic stability", studied by Mumford, Gieseker etc around 1970s.

Recently a version appeared, so-called "K-stability", introduced by

Tian(1997) and reformulated by Donaldson(2002), by the way of seeking

the analogue of Kobayashi-Hitchin correspondence, which gives

"differential geometric" interpretation of "stability". These two have

subtle but interesting differences in dimension higher than 1.

Contents:

(1*) Any semistable (in any sense) polarized variety should have only

"semi-log-canonical" singularities. (Partly observed around 1970s)

(2) On the other hand, we proved some stabilities, which corresponds to

"Calabi conjecture", also with admitting mild singularities.

As applications these yield

(3*) Compact moduli spaces with GIT interpretations.

(4) Many counterexamples (as orbifolds) to folklore conjecture:

"K-stability implies asymptotic stability".

(*: Some technical points are yet to be settled.

Some parts for (1)(2) are available on arXiv:0910.1794.)

Background:

Original GIT-stability notion for polarized variety is

"asymptotic stability", studied by Mumford, Gieseker etc around 1970s.

Recently a version appeared, so-called "K-stability", introduced by

Tian(1997) and reformulated by Donaldson(2002), by the way of seeking

the analogue of Kobayashi-Hitchin correspondence, which gives

"differential geometric" interpretation of "stability". These two have

subtle but interesting differences in dimension higher than 1.

Contents:

(1*) Any semistable (in any sense) polarized variety should have only

"semi-log-canonical" singularities. (Partly observed around 1970s)

(2) On the other hand, we proved some stabilities, which corresponds to

"Calabi conjecture", also with admitting mild singularities.

As applications these yield

(3*) Compact moduli spaces with GIT interpretations.

(4) Many counterexamples (as orbifolds) to folklore conjecture:

"K-stability implies asymptotic stability".

(*: Some technical points are yet to be settled.

Some parts for (1)(2) are available on arXiv:0910.1794.)

### 2010/05/10

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Toric degenerations of Grassmannians and mirror symmetry (JAPANESE)

**Makoto Miura**

(The University of Tokyo)Toric degenerations of Grassmannians and mirror symmetry (JAPANESE)

[ Abstract ]

I will talk about toric degenerarions of Grassmannians and

an application to the mirror constructions for complete intersection

Calabi-Yau manifolds in Grassmannians.

In particular, if we focus on toric degenerations by term orderings on

polynomial rings,

we have to choose a term ordering for which the coordinate ring has an

uniformly homogeneous sagbi basis.

We discuss this condition for some examples of ordinary Grassmannians

and a spinor variety.

I will talk about toric degenerarions of Grassmannians and

an application to the mirror constructions for complete intersection

Calabi-Yau manifolds in Grassmannians.

In particular, if we focus on toric degenerations by term orderings on

polynomial rings,

we have to choose a term ordering for which the coordinate ring has an

uniformly homogeneous sagbi basis.

We discuss this condition for some examples of ordinary Grassmannians

and a spinor variety.

### 2010/04/26

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

The unirationality of the moduli spaces of 2-elementary K3

surfaces (JAPANESE)

**Shouhei Ma**(The University of Tokyo)The unirationality of the moduli spaces of 2-elementary K3

surfaces (JAPANESE)

[ Abstract ]

We prove the unirationality of the moduli spaces of K3 surfaces

with non-symplectic involution. As a by-product, we describe the

configuration spaces of 5, 6, 7, 8 points in the projective plane as

arithmetic quotients of type IV.

We prove the unirationality of the moduli spaces of K3 surfaces

with non-symplectic involution. As a by-product, we describe the

configuration spaces of 5, 6, 7, 8 points in the projective plane as

arithmetic quotients of type IV.

### 2010/04/19

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

制限型体積と因子的ザリスキー分解

**松村 慎一**(東大数理)制限型体積と因子的ザリスキー分解

[ Abstract ]

豊富な因子の部分多様体に沿った自己交点数は, 基本的かつ重要である.

(部分多様体に沿った)自己交点数の巨大な因子への一般化である制限型体積は,

多くの状況で出現する重要な概念である.

様々な部分多様体に沿った制限型体積の振る舞いと

巨大な因子のザリスキー分解可能性の関係について考察したい.

また, 時間が許せば, 元々の問題意識であった制限型体積の複素解析的な側面に

ついても触れたい.

豊富な因子の部分多様体に沿った自己交点数は, 基本的かつ重要である.

(部分多様体に沿った)自己交点数の巨大な因子への一般化である制限型体積は,

多くの状況で出現する重要な概念である.

様々な部分多様体に沿った制限型体積の振る舞いと

巨大な因子のザリスキー分解可能性の関係について考察したい.

また, 時間が許せば, 元々の問題意識であった制限型体積の複素解析的な側面に

ついても触れたい.

### 2010/04/05

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

From Lang's Conjecture to finiteness properties of Torelli groups

**Alexandru Dimca**(Université Nice-Sophia Antipolis)From Lang's Conjecture to finiteness properties of Torelli groups

[ Abstract ]

First we recall one of Lang's conjectures in diophantine geometry

on the interplay between subvarieties and translated subgroups in a

commutative algebraic group

(proved by M. Laurent in the case of affine tori in 1984).

Then we present the technique of resonance and characteristic varieties,

a powerful tool in the study of fundamental groups of algebraic varieties.

Finally, using the two ingredients above, we show that the Torelli

groups $T_g$

have some surprising finiteness properties for $g>3$.

In particular, we show that for any subgroup $N$ in $T_g$ containing

the Johnson kernel $K_g$, the complex vector space $N_{ab} \\otimes C$

is finite dimensional.

All the details are available in our joint preprint with S. Papadima

arXiv:1002.0673.

First we recall one of Lang's conjectures in diophantine geometry

on the interplay between subvarieties and translated subgroups in a

commutative algebraic group

(proved by M. Laurent in the case of affine tori in 1984).

Then we present the technique of resonance and characteristic varieties,

a powerful tool in the study of fundamental groups of algebraic varieties.

Finally, using the two ingredients above, we show that the Torelli

groups $T_g$

have some surprising finiteness properties for $g>3$.

In particular, we show that for any subgroup $N$ in $T_g$ containing

the Johnson kernel $K_g$, the complex vector space $N_{ab} \\otimes C$

is finite dimensional.

All the details are available in our joint preprint with S. Papadima

arXiv:1002.0673.

### 2010/02/01

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Extensions of two Chow stability criteria to positive characteristics

**大川 新之介**(東大数理)Extensions of two Chow stability criteria to positive characteristics

[ Abstract ]

I will talk about two results on Chow (semi-)stability of cycles in positive characteristics, which were originally known in characteristic 0. One is on the stability of non-singular projective hypersurfaces of degree greater than 2, and the other is the criterion by Y. Lee in terms of the log canonical threshold of Chow divisor. A couple of examples will be discussed in detail.

I will talk about two results on Chow (semi-)stability of cycles in positive characteristics, which were originally known in characteristic 0. One is on the stability of non-singular projective hypersurfaces of degree greater than 2, and the other is the criterion by Y. Lee in terms of the log canonical threshold of Chow divisor. A couple of examples will be discussed in detail.

### 2010/01/25

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

On weak Fano varieties with log canonical singularities

**權業 善範**(東大数理)On weak Fano varieties with log canonical singularities

[ Abstract ]

We prove that the anti-canonical divisors of weak Fano

3-folds with log canonical singularities are semiample. Moreover, we consider

semiampleness of the anti-log canonical divisor of any weak log Fano pair

with log canonical singularities. We show semiampleness dose not hold in

general by constructing several examples. Based on those examples, we propose

sufficient conditions which seem to be the best possible and we prove

semiampleness under such conditions. In particular we derive semiampleness of the

anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers

are at most 1-dimensional. We also investigate the Kleiman-Mori cones of

weak log Fano pairs with log canonical singularities.

We prove that the anti-canonical divisors of weak Fano

3-folds with log canonical singularities are semiample. Moreover, we consider

semiampleness of the anti-log canonical divisor of any weak log Fano pair

with log canonical singularities. We show semiampleness dose not hold in

general by constructing several examples. Based on those examples, we propose

sufficient conditions which seem to be the best possible and we prove

semiampleness under such conditions. In particular we derive semiampleness of the

anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers

are at most 1-dimensional. We also investigate the Kleiman-Mori cones of

weak log Fano pairs with log canonical singularities.

### 2010/01/18

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

The divisor class group of terminal Gorenstein Fano 3-folds and rationality questions

**Anne-Sophie Kaloghiros**(RIMS)The divisor class group of terminal Gorenstein Fano 3-folds and rationality questions

[ Abstract ]

Let Y be a quartic hypersurface in CP^4 with mild singularities, e.g. no worse than ordinary double points.

If Y contains a surface that is not a hyperplane section, Y is not Q-factorial and the divisor class group of Y, Cl Y, contains divisors that are not Cartier. However, the rank of Cl Y is bounded.

In this talk, I will show that in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, the generators of Cl Y/ Pic Y are ``topological traces " of K-negative extremal contractions on X.

This has surprising consequences: it is possible to conclude that a number of families of non-factorial quartic 3-folds are rational.

In particular, I give some examples of rational quartic hypersurfaces Y_4\\subset CP^4 with rk Cl Y=2 and show that when the divisor class group of Y has sufficiently high rank, Y is always rational.

Let Y be a quartic hypersurface in CP^4 with mild singularities, e.g. no worse than ordinary double points.

If Y contains a surface that is not a hyperplane section, Y is not Q-factorial and the divisor class group of Y, Cl Y, contains divisors that are not Cartier. However, the rank of Cl Y is bounded.

In this talk, I will show that in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, the generators of Cl Y/ Pic Y are ``topological traces " of K-negative extremal contractions on X.

This has surprising consequences: it is possible to conclude that a number of families of non-factorial quartic 3-folds are rational.

In particular, I give some examples of rational quartic hypersurfaces Y_4\\subset CP^4 with rk Cl Y=2 and show that when the divisor class group of Y has sufficiently high rank, Y is always rational.

### 2009/12/21

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Ampleness of two-sided tilting complexes

**源 泰幸**(京都大学理学部数学教室)Ampleness of two-sided tilting complexes

[ Abstract ]

From the view point of noncommutative algebraic geometry (NCAG),

a two-sided tilting complex is an analog of a line bundle.

In this talk we introduce the notion of ampleness for two-sided

tilting complexes over finite dimensional algebras.

From the view point of NCAG, the Serre functors are considered to be

shifted canonical bundles. We show by examples that the property

of shifted canonical bundle captures some representation theoretic

property of algebras.

From the view point of noncommutative algebraic geometry (NCAG),

a two-sided tilting complex is an analog of a line bundle.

In this talk we introduce the notion of ampleness for two-sided

tilting complexes over finite dimensional algebras.

From the view point of NCAG, the Serre functors are considered to be

shifted canonical bundles. We show by examples that the property

of shifted canonical bundle captures some representation theoretic

property of algebras.

### 2009/12/14

14:40-16:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Invariants of Fano varieties via quantum D-module

**Sergey Galkin**(IPMU)Invariants of Fano varieties via quantum D-module

[ Abstract ]

We will introduce and compute Apery characteristic

class and Frobenius genera - invariants of Fano variety derived from

it's Gromov-Witten invariants. Then we will show how to compute them

and relate with other invariants.

We will introduce and compute Apery characteristic

class and Frobenius genera - invariants of Fano variety derived from

it's Gromov-Witten invariants. Then we will show how to compute them

and relate with other invariants.

### 2009/11/16

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Rationality of the Brauer-Severi Varieties of Skylanin algebras

**Colin Ingalls**(University of New Brunswick and RIMS)Rationality of the Brauer-Severi Varieties of Skylanin algebras

[ Abstract ]

Iskovskih's conjecture states that a conic bundle over

a surface is rational if and only if the surface has a pencil of

rational curves which meet the discriminant in 3 or fewer points,

(with one exceptional case). We generalize Iskovskih's proof that

such conic bundles are rational, to the case of projective space

bundles of higher dimension. The proof involves maximal orders

and toric geometry. As a corollary we show that the Brauer-Severi

variety of a Sklyanin algebra is rational.

Iskovskih's conjecture states that a conic bundle over

a surface is rational if and only if the surface has a pencil of

rational curves which meet the discriminant in 3 or fewer points,

(with one exceptional case). We generalize Iskovskih's proof that

such conic bundles are rational, to the case of projective space

bundles of higher dimension. The proof involves maximal orders

and toric geometry. As a corollary we show that the Brauer-Severi

variety of a Sklyanin algebra is rational.

### 2009/11/02

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Cohomology of moduli spaces of curves and modular forms

**Gerard van der Geer**(Universiteit van Amsterdam)Cohomology of moduli spaces of curves and modular forms

[ Abstract ]

The Eichler-Shimura theorem expresses cohomology of local systems

on the moduli of elliptic curves in terms of modular forms. The

cohomology of local systems can be succesfully explored by counting

points over finite fields. We show how this can be applied to

obtain a lot of information about the cohomology of other moduli spaces

of low genera and also about Siegel modular forms of genus 2 and 3.

This is joint work with Jonas Bergstroem and Carel Faber.

The Eichler-Shimura theorem expresses cohomology of local systems

on the moduli of elliptic curves in terms of modular forms. The

cohomology of local systems can be succesfully explored by counting

points over finite fields. We show how this can be applied to

obtain a lot of information about the cohomology of other moduli spaces

of low genera and also about Siegel modular forms of genus 2 and 3.

This is joint work with Jonas Bergstroem and Carel Faber.