## 代数幾何学セミナー

開催情報 火曜日　10:30～11:30 or 12:00　数理科学研究科棟(駒場) ハイブリッド開催/002号室 權業 善範・中村 勇哉・田中公

### 2010年12月13日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Sergey Fomin 氏 (University of Michigan)
Enumeration of plane curves and labeled floor diagrams (ENGLISH)
[ 講演概要 ]
Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle and G. Mikhalkin. Tropical geometry arguments yield combinatorial descriptions of (ordinary and relative) Gromov-Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In the case of the projective plane, these descriptions can be used to obtain new formulas for the corresponding enumerative invariants. In particular, we give a proof of Goettsche's polynomiality conjecture for plane curves, and enumerate plane rational curves of given degree passing through given points and having maximal tangency to a given line. On the combinatorial side, we show that labeled floor diagrams of genus 0 are equinumerous to labeled trees, and therefore counted by the celebrated Cayley's formula. The corresponding bijections lead to interpretations of the Kontsevich numbers (the genus-0 Gromov-Witten invariants of the projective plane) in terms of certain statistics on trees.

This is joint work with Grisha Mikhalkin.

### 2010年11月29日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

K3 surfaces and log del Pezzo surfaces of index three (JAPANESE)
[ 講演概要 ]
Alexeev and Nikulin have classified log del Pezzo surfaces of index 1 and 2 by using the classification of non-symplectic involutions on K3 surfaces. We want to discuss the generalization of this result to the index 3 cases. In this case we are also able to construct log del Pezzos $Z$ from K3 surfaces $X$, but the converse is not necessarily true. The condition on $Z$ is exactly the "multiple smooth divisor property", which we will define. Our theorem is the classification of log del Pezzo surfaces of index 3 with this property.

The idea of the proof is similar to that of Alexeev and Nikulin, but the methods are different because of the existence of singularities: although the singularity is mild, the description of nef cone by reflection groups cannot be used. Instead
we construct and analyze good elliptic fibrations on K3 surfaces $X$ and use it to obtain the classification. It includes a partial but geometric generalization of the classification of non-symplectic automorphisms of order three, recently done by Artebani, Sarti and Taki.

### 2010年11月16日(火)

16:30-18:00   数理科学研究科棟(駒場) 122号室
いつもと曜日・時間・場所が異なります
Viacheslav Nikulin 氏 (Univ Liverpool and Steklov Moscow)
Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)
[ 講演概要 ]
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年11月16日(火)

16:30-18:00   数理科学研究科棟(駒場) 122号室
Viacheslav Nikulin 氏 (Univ Liverpool and Steklov Moscow)
Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)
[ 講演概要 ]
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年11月15日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

Generators of tropical modules (JAPANESE)
[ 講演概要 ]
We study polytopes in a tropical projective space $X$. By Joswig and Kulas, a real convex polytope in $X$ is a tropical simplex, and therefore it is the tropically convex hull of at most $n+1$ points. We show a generalization of this result. It is given using tropical modules and its dual modules. The main interest is
the number of generators of a tropical module.

### 2010年11月01日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

How to estimate Seshadri constants (JAPANESE)
[ 講演概要 ]
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年10月18日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

ガロア拡大と局所コホモロジー間の写像について (JAPANESE)
[ 講演概要 ]

### 2010年09月06日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Prof. Remke Kloosterman 氏 (Humboldt University, Berlin)
Non-reduced components of the Noether-Lefschetz locus (ENGLISH)
[ 講演概要 ]
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年07月29日(木)

14:30-16:00   数理科学研究科棟(駒場) 126号室
いつもと曜日・時間帯が異なります。ご注意ください。

Homological Mirror Symmetry for 2-dimensional toric Fano stacks (JAPANESE)
[ 講演概要 ]
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年07月12日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

Flips of moduli of stable torsion free sheaves with $c_1=1$ on
$\\\\mathbb{P}^2$ (JAPANESE)
[ 講演概要 ]
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年07月05日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

Rational curves on hypersurfaces (JAPANESE)
[ 講演概要 ]
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年06月21日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

ファノ多様体の擬指数と端射線の長さの最小値 (JAPANESE)
[ 講演概要 ]
ファノ多様体上の有理曲線と反標準因子の交点数
の最小値は擬指数(pseudo-index)と呼ばれる。ファノ多様体
の構造は端射線によって制御されるという観点から、

### 2010年06月14日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Yongnam Lee 氏 (Sogang University)
Slope of smooth rational curves in an anticanonically polarized Fano manifold (ENGLISH)
[ 講演概要 ]
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年06月07日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Xavier Roulleau 氏 (東大数理)
Genus 2 curve configurations on Fano surfaces (ENGLISH)

### 2010年05月31日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

On Pfaffian Calabi-Yau Varieties and Mirror Symmetry (JAPANESE)
[ 講演概要 ]
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年05月24日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

A counterexample of the birational Torelli problem via Fourier--Mukai transforms (JAPANESE)
[ 講演概要 ]
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年05月17日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

On the GIT stability of Polarized Varieties (JAPANESE)
[ 講演概要 ]
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年05月10日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

Grassmann多様体のトーリック退化とミラー対称性 (JAPANESE)
[ 講演概要 ]
Grassmann多様体のトーリック退化と、
それを用いたGrassmann多様体の完全交叉カラビ・ヤウ多様体に対するミラー構

とくに、項順序によるトーリック退化に着目すれば、

A型Grassmann多様体やスピノル多様体などの例に関してこの条件を考察する。

### 2010年04月26日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

The unirationality of the moduli spaces of 2-elementary K3
surfaces (JAPANESE)
[ 講演概要 ]
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年04月19日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

[ 講演概要 ]

(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,

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年04月05日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Alexandru Dimca 氏 (Université Nice-Sophia Antipolis)
From Lang's Conjecture to finiteness properties of Torelli groups
[ 講演概要 ]
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年02月01日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

Extensions of two Chow stability criteria to positive characteristics
[ 講演概要 ]
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年01月25日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

On weak Fano varieties with log canonical singularities
[ 講演概要 ]
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年01月18日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Anne-Sophie Kaloghiros 氏 (RIMS)
The divisor class group of terminal Gorenstein Fano 3-folds and rationality questions
[ 講演概要 ]
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年12月21日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

Ampleness of two-sided tilting complexes
[ 講演概要 ]
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.