## 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

### 2011年10月31日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Minimal model theory for log surfaces (JAPANESE)
[ 講演概要 ]
We discuss the log minimal model theory for log sur- faces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.

### 2011年07月04日(月)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Birational Geometry of O'Grady's six dimensional example over the Donaldson-Uhlenbeck compactification (JAPANESE)
[ 講演概要 ]
O'Grady constructed two sporadic examples of compact irreducible symplectic Kaehler manifold, by resolving singular moduli spaces of sheaves on a K3 surface or an abelian surface. We will give a full description of the birational geometry of O'Grady's six dimensional example over the corresponding Donaldson-Uhlenbeck compactification, using an explicit calculation of certain kind of GIT quotients.
If time permits, we will also discuss an involution of the example induced by a Fourier-Mukai transformation.

### 2011年06月27日(月)

16:30-18:00   数理科学研究科棟(駒場) 126号室
Vladimir Lazić 氏 (Imperial College London)
MMP revisited, II (ENGLISH)
[ 講演概要 ]
I will talk about how finite generation of certain adjoint rings implies everything we currently know about the MMP. This is joint work with A. Corti.

### 2011年06月07日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室
Chenyang Xu 氏 (MIT)
Log canonical closure (ENGLISH)
[ 講演概要 ]
(joint with Christopher Hacon) In this talk, we will address the problem on given a log canonical variety, how we compactify it. Our approach is via MMP. The result has a few applications. Especially I will explain the one on the moduli of stable schemes.
If time permits, I will also talk about how a similar approach can be applied to give a proof of the existence of log canonical flips and a conjecture due to Kollár on the geometry of log centers.

### 2011年06月06日(月)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Multiplier ideals via Mather discrepancies (JAPANESE)
[ 講演概要 ]
For an arbitrary variety we define a multiplier ideal by using Mather discrepancy.
This ideal coincides with the usual multiplier ideal if the variety is normal and complete intersection.
In the talk I will show a local vanishing theorem for this ideal and as corollaries we obtain restriction theorem, subadditivity theorem, Skoda type theorem, and Briancon-Skoda type theorem.

### 2011年05月30日(月)

16:30-18:00   数理科学研究科棟(駒場) 126号室
Jungkai Alfred Chen 氏 (National Taiwan University and RIMS)
Kodaira Dimension of Irregular Varieties (ENGLISH)
[ 講演概要 ]
$f:X\\to Y$ be an algebraic fiber space with generic geometric fiber $F$, $\\dim X=n$ and $\\dim Y=m$. Then Iitaka's $C_{n,m}$ conjecture states $$\\kappa (X)\\geq \\kappa (Y)+\\kappa (F).$$ In particular, if $X$ is a variety with $\\kappa(X)=0$ and $f: X \\to Y$ is the Albanese map, then Ueno conjecture that $\\kappa(F)=0$. One can regard Ueno’s conjecture an important test case of Iitaka’s conjecture in general.

These conjectures are of fundamental importance in the classification of higher dimensional complex projective varieties. In a recent joint work with Hacon, we are able to prove Ueno’s conjecture and $C_{n,m}$ conjecture holds when $Y$ is of maximal Albanese dimension. In this talk, we will introduce some relative results and briefly sketch the proof.

### 2011年05月23日(月)

17:00-18:30   数理科学研究科棟(駒場) 126号室

Alpha invariant and K-stability of Fano varieties (JAPANESE)
[ 講演概要 ]
From the results of Tian, it is proved that the lower bounds of alpha invariant implies K-stability of Fano manifolds via the existence of Kähler-Einstein metrics. In this talk, I will give a direct proof of this relation in algebro-geometric way without using Kähler-Einstein metrics. This is joint work with Yuji Odaka (RIMS).

### 2011年05月16日(月)

17:00-18:30   数理科学研究科棟(駒場) 126号室

On images of Mori dream spaces (JAPANESE)
[ 講演概要 ]
Mori dream space (MDS), introduced by Y. Hu and S. Keel, is a class of varieties whose geometry can be controlled via the VGIT of the Cox ring. It is a generalization of both toric varieties and log Fano varieties.

The purpose of this talk is to study the image of a morphism from a MDS.
Firstly I prove that such an image again is a MDS.
Secondly I introduce a fan structure on the effective cone of a MDS and show that the fan of the image coincides with the restriction of that of the source.

This fan encodes some information of the Zariski decompositions, which turns out to be equivalent to the information of the GIT equivalence. In toric case, this fan coincides with the so called GKZ decomposition.

The point is that these results can be clearly explained via the VGIT description for MDS.

If I have time, I touch on generalizations and an application to the Shokurov polytopes.

### 2011年05月09日(月)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Fourier--Mukai partners of elliptic ruled surfaces (JAPANESE)
[ 講演概要 ]
Atiyah classifies vector bundles on elliptic curves E over an algebraically closed field of any characteristic. On the other hand, a rank 2 vector bundle on E defines a surface S with P^1-bundle structure on E.
We study when S has an elliptic fibration according to the Atiyah's classification. As its application, we determines the set of Fourier--Mukai partners of elliptic ruled surfaces over the complex number field.

### 2011年05月02日(月)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Projective varieties admitting an embedding with Gauss map of rank zero (JAPANESE)
[ 講演概要 ]

「ある埋込み $¥iota: X ¥hookrightarrow ¥mathbb{P}^M$ が存在し,そのガウス写像 $X ¥dashrightarrow G(¥dim(X), ¥mathbb{P}^M)$ の一般点での階数が零となる.」

### 2011年04月25日(月)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Mirror symmetry and projective geometry of Reye congruences (JAPANESE)
[ 講演概要 ]
This is a joint work with Shinobu Hosono.
It is well-known that the projective dual of the second Veronese variety v_2(P^n) is the symmetric determinantal hypersurface H. However, in the context of homological projective duality after Kuznetsov, it is natural to consider that the Chow^2 P^n and H are dual (note that Chow^2 P^n is the secant variety of v_2(P^n)).
Though we did not yet formulate what this duality exactly means in full generality, we show some results in this context for the values n¥leq 4.
For example, let n=4. We consider Chow^2 P^4 in P(S^2 V) and H in P(S^2 V^*), where V is the vector space such that P^4 =P(V). Take a general 4-plane P in
P(S^2 V^*) and let P' be the orthogonal space to P in P(S^2 V). Then X:=Chow^2 P^4 ¥cap P' is a smooth Calabi-Yau 3-fold, and there exists a natural double cover Y -> H¥cap P with a smooth Calabi-Yau 3-fold Y. It is easy to check
that X and Y are not birational each other.
Our main result asserts the derived equivalence of X and Y. This derived equivalence is given by the Fourier Mukai functor D(X)-> D(Y) whose kernel is the ideal sheaf in X×Y of a flat family of curves on Y parameterized by X.
Curves on Y in this family have degree 5 and arithmetic genus 3, and these have a nice interpretation by a BPS number of Y. The proof of the derived equivalence is slightly involved so I explain a similar result in the case where n=3. In this case, we obtain a fully faithful functor from D(X)-> D(Y), where X is a so called the Reye congruence Enriques surface and Y is the 'big resolution' of the Artin-Mumford quartic double solid.

### 2011年04月18日(月)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Ideal-adic semi-continuity problem for minimal log discrepancies (JAPANESE)
[ 講演概要 ]
De Fernex, Ein and Mustaţă, after Kollár, proved the ideal-adic semi-continuity of log canonicity to obtain Shokurov's ACC conjecture for log canonical thresholds on l.c.i. varieties. I discuss its generalisation to minimal log discrepancies, proposed by Mustaţă.

### 2011年01月31日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Sukmoon Huh 氏 (KIAS)
Restriction maps to the Coble quartic (ENGLISH)
[ 講演概要 ]
The Coble sixfold quartic is the moduli space of semi-stable vector bundle of rank 2 on a non-hyperelliptic curve of genus 3 with canonical determinant. Considering the curve as a plane quartic, we investigate the restriction of the semi-stable sheaves over the projective plane to the curve. We suggest a positive side of this trick in the study of the moduli space of vector bundles over curves by showing several examples such as Brill-Noether loci and a few rational subvarieties of the Coble quartic. In a later part of the talk, we introduce the rationality problem of the Coble quartic. If the time permits, we will apply the same idea to the moduli space of bundles over curves of genus 4 to derive some geometric properties of the Brill-Noether loci in the case of genus 4.

### 2011年01月17日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室
Dano Kim 氏 (KIAS)
L^2 methods and Skoda division theorems (ENGLISH)
[ 講演概要 ]
Extension of Ohsawa-Takegoshi type and division of Skoda type are two important consequences of the L^2 methods of Hormander, Demailly and others. They are analogous to vanishing theorems of Kodaira type and can be viewed as some refinement of the vanishing. The best illustration of their usefulness up to now is Siu’s proof of invariance of plurigenera without general type assumption. In this talk, we will focus on the division theorem / problem and talk about its currently known cases (old and new). One motivation comes from yet another viewpoint on the finite generation of canonical ring.

### 2010年12月20日(月)

16:40-18:10   数理科学研究科棟(駒場) 126号室

On the minimal model theory from a viewpoint of numerical invariants (JAPANESE)
[ 講演概要 ]
I will introduce the numerical Kodaira dimension for pseudo-effective divisors after N. Nakayama and explain the minimal model theory of numerical Kodaira dimension zero. I also will talk about the applications. ( partially joint work with B. Lehmann.)

### 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.