## 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**

### 2015/11/16

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

Counting curves on surface in Calabi-Yau threefolds and the proof of S-duality modularity conjecture (English)

**Artan Sheshmani**(IPMU/ Ohio State University)Counting curves on surface in Calabi-Yau threefolds and the proof of S-duality modularity conjecture (English)

[ Abstract ]

I will talk about recent joint works with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve a complete algebraic-geometric proof of S-duality modularity conjecture.

I will talk about recent joint works with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve a complete algebraic-geometric proof of S-duality modularity conjecture.

### 2015/11/09

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

3-dimensional McKay correspondence (English)

**Yukari Ito**(Nagoya University)3-dimensional McKay correspondence (English)

[ Abstract ]

The original McKay correspondence is a relation between group theory of a finite subgroup G of SL(2,C) and geometry of the minimal resolution of the quotient singularity by G, and was generalized several ways. In particular, 3-dimensional generalization was extended to derived categorical eqivalence and the G-Hilbert scheme was useful to explain the correspondence. However, most results hold only for abelian subgroups. In this talk, I would like to introduce an iterated G-Hilbert scheme and show more geometrical McKay correspondence for non-abelian subgroups.

The original McKay correspondence is a relation between group theory of a finite subgroup G of SL(2,C) and geometry of the minimal resolution of the quotient singularity by G, and was generalized several ways. In particular, 3-dimensional generalization was extended to derived categorical eqivalence and the G-Hilbert scheme was useful to explain the correspondence. However, most results hold only for abelian subgroups. In this talk, I would like to introduce an iterated G-Hilbert scheme and show more geometrical McKay correspondence for non-abelian subgroups.

### 2015/11/05

15:30-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Compact moduli of marked noncommutative del Pezzo surfaces via quivers (English)

**Shinnosuke Okawa**(Osaka University)Compact moduli of marked noncommutative del Pezzo surfaces via quivers (English)

[ Abstract ]

I will introduce certain GIT construction via quivers of compactified moduli spaces of marked noncommutative del Pezzo surfaces. For projective plane, quadric surface, and those of degree 3, 2, 1, we obtain projective toric varieties of dimension 2, 3, 8, 9, 10, respectively. Then I will discuss relations with deformation theory of abelian categories, blow-up of noncommutative projective planes, and three-block exceptional collections due to Karpov and Nogin. This talk is based on joint works in progress with Tarig Abdelgadir and Kazushi Ueda.

I will introduce certain GIT construction via quivers of compactified moduli spaces of marked noncommutative del Pezzo surfaces. For projective plane, quadric surface, and those of degree 3, 2, 1, we obtain projective toric varieties of dimension 2, 3, 8, 9, 10, respectively. Then I will discuss relations with deformation theory of abelian categories, blow-up of noncommutative projective planes, and three-block exceptional collections due to Karpov and Nogin. This talk is based on joint works in progress with Tarig Abdelgadir and Kazushi Ueda.

### 2015/10/26

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

Asymptotic syzygies and the gonality conjecture (English)

**Lawrence Ein**(University of Illinois at Chicago)Asymptotic syzygies and the gonality conjecture (English)

[ Abstract ]

We'll discuss my joint work with Lazarsfeld on the gonality conjecture about the syzygies of a smooth projective curve when it is embedded into the projective space by the complete linear system of a sufficiently very ample line bundles. We'll also discuss some results about the asymptotic syzygies f higher dimensional varieties.

We'll discuss my joint work with Lazarsfeld on the gonality conjecture about the syzygies of a smooth projective curve when it is embedded into the projective space by the complete linear system of a sufficiently very ample line bundles. We'll also discuss some results about the asymptotic syzygies f higher dimensional varieties.

### 2015/10/05

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

Weighted Compactifications of Configuration Spaces (English)

**Evangelos Routis**(IPMU)Weighted Compactifications of Configuration Spaces (English)

[ Abstract ]

In the early 90's, Fulton and MacPherson provided a natural and beautiful way of compactifying the configuration space F(X,n) of n distinct labeled points on an arbitrary nonsingular variety. In this talk, I will present an alternate compactification of F(X,n), which generalizes the work of Fulton and MacPherson and is parallel to Hassett's weighted generalization of the moduli space of n-pointed stable curves. After discussing its main properties, I will give a presentation of its intersection ring and as an application, I will describe the intersection ring of Hassett's spaces in genus 0. Finally, as time permits, I will discuss some additional moduli problems associated with weighted compactifications.

In the early 90's, Fulton and MacPherson provided a natural and beautiful way of compactifying the configuration space F(X,n) of n distinct labeled points on an arbitrary nonsingular variety. In this talk, I will present an alternate compactification of F(X,n), which generalizes the work of Fulton and MacPherson and is parallel to Hassett's weighted generalization of the moduli space of n-pointed stable curves. After discussing its main properties, I will give a presentation of its intersection ring and as an application, I will describe the intersection ring of Hassett's spaces in genus 0. Finally, as time permits, I will discuss some additional moduli problems associated with weighted compactifications.

### 2015/06/29

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

Twisted cubics and cubic fourfolds (English)

**Manfred Lehn**(Mainz/RIMS)Twisted cubics and cubic fourfolds (English)

[ Abstract ]

The moduli scheme of generalised twisted cubics on a smooth

cubic fourfold Y non containing a plane is smooth projective of

dimension 10 and admits a contraction to an 8-dimensional

holomorphic symplectic manifold Z(Y). The latter is shown to be

birational to the Hilbert scheme of four points on a K3 surface if

Y is of Pfaffian type. This is a report on joint work with C. Lehn,

C. Sorger and D. van Straten and with N. Addington.

The moduli scheme of generalised twisted cubics on a smooth

cubic fourfold Y non containing a plane is smooth projective of

dimension 10 and admits a contraction to an 8-dimensional

holomorphic symplectic manifold Z(Y). The latter is shown to be

birational to the Hilbert scheme of four points on a K3 surface if

Y is of Pfaffian type. This is a report on joint work with C. Lehn,

C. Sorger and D. van Straten and with N. Addington.

### 2015/06/22

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

Rational cohomology tori

(English)

http://webusers.imj-prg.fr/~marti.lahoz/

**Martí Lahoz**(Institut de Mathématiques de Jussieu )Rational cohomology tori

(English)

[ Abstract ]

Complex tori can be topologically characterised among compact Kähler

manifolds by their integral cohomology ring. I will discuss the

structure of compact Kähler manifolds whose rational cohomology ring is

isomorphic to the rational cohomology ring of a torus and give some

examples. This is joint work with Olivier Debarre and Zhi Jiang.

[ Reference URL ]Complex tori can be topologically characterised among compact Kähler

manifolds by their integral cohomology ring. I will discuss the

structure of compact Kähler manifolds whose rational cohomology ring is

isomorphic to the rational cohomology ring of a torus and give some

examples. This is joint work with Olivier Debarre and Zhi Jiang.

http://webusers.imj-prg.fr/~marti.lahoz/

### 2015/06/15

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

Boundedness of the KSBA functor of

SLC models (English)

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

**Christopher Hacon**(University of Utah/RIMS)Boundedness of the KSBA functor of

SLC models (English)

[ Abstract ]

Let $X$ be a canonically polarized smooth $n$-dimensional projective variety over $\mathbb C$ (so that $\omega _X$ is ample), then it is well-known that a fixed multiple of the canonical line bundle defines an embedding of $X$ in projective space. It then follows easily that if we fix certain invariants of $X$, then $X$ belongs to finitely many deformation types. Since canonical models are rarely smooth, it is important to generalize this result to canonically polarized $n$-dimensional projectivevarieties with canonical singularities. Moreover, since these varieties specialize to non-normal varieties it is also important to generalize this result to semi-log canonical pairs. In this talk we will explain a strong version of the above result that applies to semi-log canonical pairs.This is joint work with C. Xu and J. McKernan

[ Reference URL ]Let $X$ be a canonically polarized smooth $n$-dimensional projective variety over $\mathbb C$ (so that $\omega _X$ is ample), then it is well-known that a fixed multiple of the canonical line bundle defines an embedding of $X$ in projective space. It then follows easily that if we fix certain invariants of $X$, then $X$ belongs to finitely many deformation types. Since canonical models are rarely smooth, it is important to generalize this result to canonically polarized $n$-dimensional projectivevarieties with canonical singularities. Moreover, since these varieties specialize to non-normal varieties it is also important to generalize this result to semi-log canonical pairs. In this talk we will explain a strong version of the above result that applies to semi-log canonical pairs.This is joint work with C. Xu and J. McKernan

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

### 2015/06/01

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

Rank 2 weak Fano bundles on cubic 3-folds (日本語)

**Daizo Ishikawa**(Waseda University)Rank 2 weak Fano bundles on cubic 3-folds (日本語)

[ Abstract ]

A vector bundle on a projective variety is called weak Fano if its

projectivization is a weak Fano manifold. This is a generalization of

Fano bundles.

In this talk, we will obtain a classification of rank 2 weak Fano

bundles on a nonsingular cubic hypersurface in a projective 4-space.

Specifically, we will show that there exist rank 2 indecomposable weak

Fano bundles on it.

A vector bundle on a projective variety is called weak Fano if its

projectivization is a weak Fano manifold. This is a generalization of

Fano bundles.

In this talk, we will obtain a classification of rank 2 weak Fano

bundles on a nonsingular cubic hypersurface in a projective 4-space.

Specifically, we will show that there exist rank 2 indecomposable weak

Fano bundles on it.

### 2015/05/25

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

Good reduction of K3 surfaces (日本語 or English)

https://www.ms.u-tokyo.ac.jp/~ymatsu/index_j.html

**Yuya Matsumoto**(University of Tokyo)Good reduction of K3 surfaces (日本語 or English)

[ Abstract ]

We consider degeneration of K3 surfaces over a 1-dimensional base scheme

of mixed characteristic (e.g. Spec of the p-adic integers).

Under the assumption of potential semistable reduction, we first prove

that a trivial monodromy action on the l-adic etale cohomology group

implies potential good reduction, where potential means that we allow a

finite base extension.

Moreover we show that a finite etale base change suffices.

The proof for the first part involves a mixed characteristic

3-dimensional MMP (Kawamata) and the classification of semistable

degeneration of K3 surfaces (Kulikov, Persson--Pinkham, Nakkajima).

For the second part, we consider flops and descent arguments. This is a joint work with Christian Liedtke.

[ Reference URL ]We consider degeneration of K3 surfaces over a 1-dimensional base scheme

of mixed characteristic (e.g. Spec of the p-adic integers).

Under the assumption of potential semistable reduction, we first prove

that a trivial monodromy action on the l-adic etale cohomology group

implies potential good reduction, where potential means that we allow a

finite base extension.

Moreover we show that a finite etale base change suffices.

The proof for the first part involves a mixed characteristic

3-dimensional MMP (Kawamata) and the classification of semistable

degeneration of K3 surfaces (Kulikov, Persson--Pinkham, Nakkajima).

For the second part, we consider flops and descent arguments. This is a joint work with Christian Liedtke.

https://www.ms.u-tokyo.ac.jp/~ymatsu/index_j.html

### 2015/05/18

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

Twists and braids for general 3-fold flops (English)

http://db.ipmu.jp/member/personal/4007en.html

**Will Donovan**(IPMU)Twists and braids for general 3-fold flops (English)

[ Abstract ]

When a 3-fold contains a floppable rational curve, a theorem of Bridgeland provides a derived equivalence between the 3-fold and its flop. I will discuss recent joint work with Michael Wemyss, showing that these flop functors satisfy Coxeter-type braid relations. Using this result, we construct an action of a braid-type group on the derived category of the 3-fold. This group arises from the topology of a certain simplicial hyperplane arrangement, determined by the local geometry of the curve. I will give examples and explain key elements in the construction, including the noncommutative deformations of curves introduced in our previous work.

[ Reference URL ]When a 3-fold contains a floppable rational curve, a theorem of Bridgeland provides a derived equivalence between the 3-fold and its flop. I will discuss recent joint work with Michael Wemyss, showing that these flop functors satisfy Coxeter-type braid relations. Using this result, we construct an action of a braid-type group on the derived category of the 3-fold. This group arises from the topology of a certain simplicial hyperplane arrangement, determined by the local geometry of the curve. I will give examples and explain key elements in the construction, including the noncommutative deformations of curves introduced in our previous work.

http://db.ipmu.jp/member/personal/4007en.html

### 2015/05/11

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

Deformations of weak Fano varieties (日本語 or English)

https://sites.google.com/site/tarosano222/

**Taro Sano**(Kyoto University)Deformations of weak Fano varieties (日本語 or English)

[ Abstract ]

A smooth projective variety often has obstructed deformations.

Nevertheless, important varieties such as Fano varieties and

Calabi-Yau varieties have unobstructed deformations.

In this talk, I explain about unobstructedness of deformations of weak

Fano varieties, in particular a weak Q-Fano 3-fold.

I also present several examples to show delicateness of this unobstructedness.

[ Reference URL ]A smooth projective variety often has obstructed deformations.

Nevertheless, important varieties such as Fano varieties and

Calabi-Yau varieties have unobstructed deformations.

In this talk, I explain about unobstructedness of deformations of weak

Fano varieties, in particular a weak Q-Fano 3-fold.

I also present several examples to show delicateness of this unobstructedness.

https://sites.google.com/site/tarosano222/

### 2015/04/27

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

Lagrangian embeddings of cubic fourfolds containing a plane (日本語)

**Genki Ouchi**(University of Tokyo/IPMU)Lagrangian embeddings of cubic fourfolds containing a plane (日本語)

### 2015/04/20

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

Fano 5-folds with nef tangent bundles (日本語)

**Akihiro Kanemitsu**(University of Tokyo)Fano 5-folds with nef tangent bundles (日本語)

### 2015/04/13

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

An orbifold version of Miyaoka's semi-positivity theorem and applications (English)

**Frédéric Campana**(Université de Lorraine)An orbifold version of Miyaoka's semi-positivity theorem and applications (English)

[ Abstract ]

This `orbifold' version of Miyaoka's theorem says that if (X,D)

is a projective log-canonical pair with K_X+D pseudo-effective,

then its 'cotangent' sheaf $¥Omega^1(X,D)$ is generically semi-positive.

The definitions will be given. The original proof of Miyaoka, which

mixes

char 0 and char p>0 arguments could not be adapted. Our proof is in char

0 only.

A first consequence is when (X,D) is log-smooth with reduced boudary D,

in which case the cotangent sheaf is the classical Log-cotangent sheaf:

if some tensor power of $¥omega^1_X(log(D))$ contains a 'big' line

bundle, then K_X+D is 'big' too. This implies, together with work of

Viehweg-Zuo,

the `hyperbolicity conjecture' of Shafarevich-Viehweg.

The preceding is joint work with Mihai Paun.

A second application (joint work with E. Amerik) shows that if D is a

non-uniruled smooth divisor in aprojective hyperkaehler manifold with

symplectic form s,

then its characteristic foliation is algebraic only if X is a K3 surface.

This was shown previously bt Hwang-Viehweg assuming D to be of general

type. This result has some further consequences.

This `orbifold' version of Miyaoka's theorem says that if (X,D)

is a projective log-canonical pair with K_X+D pseudo-effective,

then its 'cotangent' sheaf $¥Omega^1(X,D)$ is generically semi-positive.

The definitions will be given. The original proof of Miyaoka, which

mixes

char 0 and char p>0 arguments could not be adapted. Our proof is in char

0 only.

A first consequence is when (X,D) is log-smooth with reduced boudary D,

in which case the cotangent sheaf is the classical Log-cotangent sheaf:

if some tensor power of $¥omega^1_X(log(D))$ contains a 'big' line

bundle, then K_X+D is 'big' too. This implies, together with work of

Viehweg-Zuo,

the `hyperbolicity conjecture' of Shafarevich-Viehweg.

The preceding is joint work with Mihai Paun.

A second application (joint work with E. Amerik) shows that if D is a

non-uniruled smooth divisor in aprojective hyperkaehler manifold with

symplectic form s,

then its characteristic foliation is algebraic only if X is a K3 surface.

This was shown previously bt Hwang-Viehweg assuming D to be of general

type. This result has some further consequences.

### 2015/01/26

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

Positivity in varieties of maximal Albanese dimension (ENGLISH)

**Jungkai Chen**(National Taiwan University)Positivity in varieties of maximal Albanese dimension (ENGLISH)

[ Abstract ]

Given a variety of maximal Albanese dimension, it is known that the holomorphic Euler characteristic is non-negative. It is an interesting question to characterize varieties with vanishing Euler characteristic.

In our previous work (jointly with Debarre and Jiang), we prove that Ein-Lazarsgfeld's example is essentially the only variety of maximal Albanese and Kodaira dimension with vanishing Euler characteristic in dimension three. In the recent joint work with Jiang, we prove a decomposition theorem for the push-forward of canonical sheaf. As a consequence, we are able to generalized our previous characterization. The purpose of this talk is give a survey of these two works.

Given a variety of maximal Albanese dimension, it is known that the holomorphic Euler characteristic is non-negative. It is an interesting question to characterize varieties with vanishing Euler characteristic.

In our previous work (jointly with Debarre and Jiang), we prove that Ein-Lazarsgfeld's example is essentially the only variety of maximal Albanese and Kodaira dimension with vanishing Euler characteristic in dimension three. In the recent joint work with Jiang, we prove a decomposition theorem for the push-forward of canonical sheaf. As a consequence, we are able to generalized our previous characterization. The purpose of this talk is give a survey of these two works.

### 2015/01/19

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

Crepant resolutions of Slodowy slice in nilpotent orbit closure in sl_N(C) (JAPANESE)

**Ryo Yamagishi**(Kyoto University)Crepant resolutions of Slodowy slice in nilpotent orbit closure in sl_N(C) (JAPANESE)

[ Abstract ]

Nilpotent orbit closures and their intersections with Slodowy slices are typical examples of symplectic varieties. It is known that every crepant resolution of a nilpotent orbit closure is obtained as a Springer resolution. In this talk, we show that every crepant resolution of a Slodowy slice in nilpotent orbit closure in sl_N(C) is obtained as the restriction of a Springer resolution and explain how to count the number of crepant resolutions. The proof of the main results is based on the fact that Slodowy slices can be described as quiver varieties.

Nilpotent orbit closures and their intersections with Slodowy slices are typical examples of symplectic varieties. It is known that every crepant resolution of a nilpotent orbit closure is obtained as a Springer resolution. In this talk, we show that every crepant resolution of a Slodowy slice in nilpotent orbit closure in sl_N(C) is obtained as the restriction of a Springer resolution and explain how to count the number of crepant resolutions. The proof of the main results is based on the fact that Slodowy slices can be described as quiver varieties.

### 2014/12/15

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

A characterization of ordinary abelian varieties in positive characteristic (JAPANESE)

**Akiyoshi Sannai**(University of Tokyo)A characterization of ordinary abelian varieties in positive characteristic (JAPANESE)

[ Abstract ]

This is joint work with Hiromu Tanaka. In this talk, we study F^e_*O_X on a projective variety over the algebraic closed field of positive characteristic. For an ordinary abelian variety X, F^e_*O_X is decomposed into line bundles for every positive integer e. Conversely, if a smooth projective variety X satisfies this property and its Kodaira dimension is non-negative, then X is an ordinary abelian variety.

This is joint work with Hiromu Tanaka. In this talk, we study F^e_*O_X on a projective variety over the algebraic closed field of positive characteristic. For an ordinary abelian variety X, F^e_*O_X is decomposed into line bundles for every positive integer e. Conversely, if a smooth projective variety X satisfies this property and its Kodaira dimension is non-negative, then X is an ordinary abelian variety.

### 2014/12/01

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

Induced Automorphisms on Hyperkaehler Manifolds (ENGLISH)

**Malte Wandel**(RIMS)Induced Automorphisms on Hyperkaehler Manifolds (ENGLISH)

[ Abstract ]

in this talk I want to report on a joint project with Giovanni Mongardi (Milano). We study automorphisms of hyperkaehler manifolds. All known deformation classes of these manifolds contain moduli spaces of stable sheaves on surfaces. If the underlying surface admits a non-trivial automorphism, it is often possible to transfer this automorphism to a moduli space of sheaves. In this way we obtain a big class of interesting examples of automorphisms of hyperkaehler manifolds. I will present a criterion to 'detect' automorphisms in this class and discuss several applications for the classification of automorphisms of manifolds of K3^[n]- and kummer n-type. If time permits I will try to talk about generalisations to O'Grady's sporadic examples.

in this talk I want to report on a joint project with Giovanni Mongardi (Milano). We study automorphisms of hyperkaehler manifolds. All known deformation classes of these manifolds contain moduli spaces of stable sheaves on surfaces. If the underlying surface admits a non-trivial automorphism, it is often possible to transfer this automorphism to a moduli space of sheaves. In this way we obtain a big class of interesting examples of automorphisms of hyperkaehler manifolds. I will present a criterion to 'detect' automorphisms in this class and discuss several applications for the classification of automorphisms of manifolds of K3^[n]- and kummer n-type. If time permits I will try to talk about generalisations to O'Grady's sporadic examples.

### 2014/10/27

14:50-16:20 Room #122 (Graduate School of Math. Sci. Bldg.)

On projective varieties with very large canonical volume (ENGLISH)

**Meng Chen**(Fudan University)On projective varieties with very large canonical volume (ENGLISH)

[ Abstract ]

For any positive integer n>0, a theorem of Hacon-McKernan, Takayama and Tsuji says that there is a constant c(n) so that the m-canonical map is birational onto its image for all smooth projective n-folds and all m>=c(n). We are interested in the following problem "P(n)": is there a constant M(n) so that, for all smooth projective n-fold X with Vol(X)>M(n), the m-canonical map of X is birational for all m>=c(n-1). The answer to “P_n" is positive due to Bombieri when $n=2$ and to Todorov when $n=3$. The aim of this talk is to introduce my joint work with Zhi Jiang from Universite Paris-Sud. We give a positive answer in dimensions 4 and 5.

For any positive integer n>0, a theorem of Hacon-McKernan, Takayama and Tsuji says that there is a constant c(n) so that the m-canonical map is birational onto its image for all smooth projective n-folds and all m>=c(n). We are interested in the following problem "P(n)": is there a constant M(n) so that, for all smooth projective n-fold X with Vol(X)>M(n), the m-canonical map of X is birational for all m>=c(n-1). The answer to “P_n" is positive due to Bombieri when $n=2$ and to Todorov when $n=3$. The aim of this talk is to introduce my joint work with Zhi Jiang from Universite Paris-Sud. We give a positive answer in dimensions 4 and 5.

### 2014/07/07

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

Balanced line bundles (JAPANESE)

**Sho Tanimoto**(Rice University)Balanced line bundles (JAPANESE)

[ Abstract ]

A conjecture of Batyrev and Manin relates arithmetic properties of

varieties with big anticanonical class to geometric invariants; in

particular, counting functions defined by metrized ample line bundles

and the corresponding asymptotics of rational points of bounded height

are interpreted in terms of cones of effective divisors and certain

thresholds with respect to these cones. This framework leads to the

notion of balanced line bundles, whose counting functions, conjecturally,

capture generic distributions of rational points. We investigate

balanced line bundles in the context of the Minimal Model Program, with

special regard to the classification of Fano threefolds and Mori fiber

spaces.

This is joint work with Brian Lehmann and Yuri Tschinkel.

A conjecture of Batyrev and Manin relates arithmetic properties of

varieties with big anticanonical class to geometric invariants; in

particular, counting functions defined by metrized ample line bundles

and the corresponding asymptotics of rational points of bounded height

are interpreted in terms of cones of effective divisors and certain

thresholds with respect to these cones. This framework leads to the

notion of balanced line bundles, whose counting functions, conjecturally,

capture generic distributions of rational points. We investigate

balanced line bundles in the context of the Minimal Model Program, with

special regard to the classification of Fano threefolds and Mori fiber

spaces.

This is joint work with Brian Lehmann and Yuri Tschinkel.

### 2014/06/30

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

Invariant subrings of the Cox rings of K3surfaces by automorphism groups (JAPANESE)

**Akiyoshi Sannai**(University of Tokyo)Invariant subrings of the Cox rings of K3surfaces by automorphism groups (JAPANESE)

[ Abstract ]

Cox rings were introduced by D.Cox and are important rings which appeared in algebraic geometry. One of the main topic related with Cox rings is the finite generation of them. In this talk, we consider the Cox rings of K3 surfaces and answer the following question asked by D. Huybrechts; Are the invariant subrings of the Cox rings of K3 surfaces by automorphism groups finitely generated in general?

Cox rings were introduced by D.Cox and are important rings which appeared in algebraic geometry. One of the main topic related with Cox rings is the finite generation of them. In this talk, we consider the Cox rings of K3 surfaces and answer the following question asked by D. Huybrechts; Are the invariant subrings of the Cox rings of K3 surfaces by automorphism groups finitely generated in general?

### 2014/06/02

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

On base point free theorem for log canonical three folds over the algebraic closure of a finite field (JAPANESE)

**Yusuke Nakamura**(University of Tokyo)On base point free theorem for log canonical three folds over the algebraic closure of a finite field (JAPANESE)

[ Abstract ]

We will discuss about the base point free theorem on three-dimensional

pairs defined over the algebraic closure of a finite field.

We know the base point free theorem on arbitrary-dimensional Kawamata

log terminal pairs in characteristic zero. By Birkar and Xu, the base

point free theorem in positive characteristic is known for big line

bundles on three-dimensional Kawamata log terminal pairs defined over

an algebraically closed field of characteristic larger than 5. Over the

algebraic closure of a finite field, a stronger result was proved by Keel.

The purpose of this talk is to generalize the Keel's result. We will

prove the base point free theorem for big line bundles on

three-dimensional log canonical pairs defined over the algebraic closure

of a finite field. This theorem is not valid for another field.

This is joint work with Diletta Martinelli and Jakub Witaszek.

We will discuss about the base point free theorem on three-dimensional

pairs defined over the algebraic closure of a finite field.

We know the base point free theorem on arbitrary-dimensional Kawamata

log terminal pairs in characteristic zero. By Birkar and Xu, the base

point free theorem in positive characteristic is known for big line

bundles on three-dimensional Kawamata log terminal pairs defined over

an algebraically closed field of characteristic larger than 5. Over the

algebraic closure of a finite field, a stronger result was proved by Keel.

The purpose of this talk is to generalize the Keel's result. We will

prove the base point free theorem for big line bundles on

three-dimensional log canonical pairs defined over the algebraic closure

of a finite field. This theorem is not valid for another field.

This is joint work with Diletta Martinelli and Jakub Witaszek.

### 2014/05/12

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

Higher Nash blowup on normal toric varieties and a higher order version of Nobile's theorem (ENGLISH)

**Andrés Daniel Duarte**(Institut de Mathématiques de Toulouse)Higher Nash blowup on normal toric varieties and a higher order version of Nobile's theorem (ENGLISH)

[ Abstract ]

The higher Nash blowup of an algebraic variety replaces singular points with limits of certain vector spaces carrying first or higher order data associated to the variety at non-singular points. In the case of normal toric varieties, the higher Nash blowup has a combinatorial description in terms of the Gröbner fan. This description will allows us to prove a higher version of Nobile's theorem in this context: for a normal toric variety, the higher Nash blowup is an isomorphism if and only if the variety is non-singular. We will also present some further observations coming from computational experiments.

The higher Nash blowup of an algebraic variety replaces singular points with limits of certain vector spaces carrying first or higher order data associated to the variety at non-singular points. In the case of normal toric varieties, the higher Nash blowup has a combinatorial description in terms of the Gröbner fan. This description will allows us to prove a higher version of Nobile's theorem in this context: for a normal toric variety, the higher Nash blowup is an isomorphism if and only if the variety is non-singular. We will also present some further observations coming from computational experiments.

### 2014/04/28

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

Syzygies of jacobian ideals and Torelli properties (ENGLISH)

**Alexandru Dimca**(Institut Universitaire de France )Syzygies of jacobian ideals and Torelli properties (ENGLISH)

[ Abstract ]

Let $C$ be a reduced complex projective plane curve defined by a homogeneous equation $f(x,y,z)=0$. We consider syzygies of the type $af_x+bf_y+cf_z=0$, where $a,b,c$ are homogeneous polynomials and $f_x,f_y,f_z$ stand for the partial derivatives of $f$. In our talk we relate such syzygies with stable or splittable rank two vector bundles on the projective plane, and to Torelli properties of plane curves in the sense of Dolgachev-Kapranov.

Let $C$ be a reduced complex projective plane curve defined by a homogeneous equation $f(x,y,z)=0$. We consider syzygies of the type $af_x+bf_y+cf_z=0$, where $a,b,c$ are homogeneous polynomials and $f_x,f_y,f_z$ stand for the partial derivatives of $f$. In our talk we relate such syzygies with stable or splittable rank two vector bundles on the projective plane, and to Torelli properties of plane curves in the sense of Dolgachev-Kapranov.