Algebraic Geometry Seminar

Seminar information archive ~06/17Next seminarFuture seminars 06/18~

Date, time & place Friday 13:30 - 15:00 ハイブリッド開催/117Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)
Organizer(s) GONGYO Yoshinori, NAKAMURA Yusuke, TANAKA Hiromu

Seminar information archive

2009/07/13

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
佐野 太郎 (東大数理)
Seshadri constants on rational surfaces with anticanonical pencils

[ Abstract ]
射影多様体上の豊富線束の$k$-jet ample性を測る不変量として
Seshadri定数と呼ばれる正の実数がある。
この不変量を調べることでしばしば幾何的な情報が得られる。
今回、1次元以上の反標準線形系をもつ有理曲面上のSeshadri定数を計算する公式
が得られた。
その公式を使うと、対数del Pezzo曲面の特異点の情報をSeshadri定数の値から
復元できる。

2009/07/06

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
柳田 伸太郎 (神戸大学理学研究科)
アーベル曲面上の安定層とフーリエ向井変換について
[ Abstract ]
今回の講演は吉岡康太との共同研究に基づくものである. 研究の発端は, 向井茂が1980年前後(フーリエ向井変換の発見前後)に考察し, 当時の講演記録に書き残した主張や予想の解読にある.
本研究は, 大まかに言うと, 半等質層とフーリエ向井変換を用いて, アーベル曲面上の安定層のモジュライ空間の構造を調べるというものである.
アーベル曲面上には半等質層と呼ばれる半安定層があり, その分類, 構成方法やコホモロジーが完全に知られている. アーベル曲面のフーリエ向井対は半等質層のモジュライ空間であることも知られている.
今回の研究はこの半等質層をbulding blockとして一般の安定層を構成することを考える. その際に"semi-homogeneous presentation"という概念が必要になる. これはアーベル曲面上の安定層の半等質層によるある種の分解のことである. 曲面のピカール数が1の時, この種の分解の存在が安定層のチャーン指標のみを用いて判定できる.
また安定層のフーリエ変換における振舞いの記述において, 算術群や整数係数2次形式が重要な役割を果たすことも分かる. この事と先に述べた表示の存在から, 安定層のモジュライとアーベル曲面上の点のヒルベルトスキームとの間の双有理変換が明示的に構成できる.
アーベル曲面のフーリエ向井変換のフォーマリズムはK3曲面の変換と共通する部分も少なくない. 講演ではそうした点にも触れつつ, 今回の結果とその証明の概要を解説したい.

2009/06/29

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
大川 領 (東京工業大学)
Moduli on the projective plane and the wall-crossing
[ Abstract ]
射影平面上の半安定層のモジュライ空間を、Bridgeland 安定性条件
を用いることにより、ある有限次元代数の半安定表現のモジュライ空間
として構成する。階数が2以下の場合、表現の安定性条件を変化させること
により、壁越え現象としてのflip の記述を得る。
応用として、flip のBetti 数などが計算できる。

2009/06/23

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
岸本崇氏 (埼玉大学理工学研究科)
Group actions on affine cones
[ Abstract ]
The action of the additive group scheme C_+ on normal affine varieties is one of main subjects in affine algebraic geometry for a long time. In this talk, we shall mainly consider the problem about the existence of C_+-actions on affine cones, more precisely, the question:

"Determine the affine cones over smooth projective varieties admitting a (non-trivial) C_+-action ".

This question has an interest from a point of view of singularities. Indeed, a normal Cohen-Macaulay affine variety admitting an action by C_+ has at most rational singularities due to the result of H. Flenner and M. Zaidenberg. In the case of dimension 2, any affine cone over the projective line P^1 has a cyclic quotient singularity, and we can see that it admits, in fact, a C_+-action. Meanwhile, in case of dimension 3, i.e., affine cones over rational surfaces, the situation becomes more subtle.

One of the main results is concerned with a criterion for the existence of a C_+-action on affine cones (of any dimension) in terms of a cylinderlike open subset on the base variety. By making use of it, it is shown that, for any rational surface Y, we can take a suitable embedding of Y in such a way that the associated affine cone admits an action of C_+. Furthermore we are able to confirm that an affine cone over an anticanonically embedded del Pezzo surface of degree greater than or equal to 4 also admits such an action.

Nevertheless, our final purpose to decide whether or not there does exist a C_+-action on the fermat cubic: x^3+y^3+z^3+u^3 =0 in C^4, which is the affine cone over an anticanonically embedded cubic surface, say Y_3, is not yet accomplished. But, we can obtain certain informations about a linear pencil of rational curves on Y_3 arising from a C_+-action which seem to be useful in order to deny an existence of an action of C_+.

2009/06/15

16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
馬 昭平氏 (東大数理)
アーベル曲面の分解と2次形式

[ Abstract ]
複素Abel曲面が楕円曲線の積に分解可能である時、分解の仕方は一般に何通りも
ありうる。いくつかの場合に分解の個数公式が求められてきた(林田、塩田-三谷
)。本講演では、すべての分解可能な複素Abel曲面に対して、2次形式論の技法
を用いて分解数の公式を与える。関連して次のことも話す:合同モジュラー曲線
上のAtkin-Lehner対合の幾何学的意味;正定値2元2次形式の類数と判別式形式
の等長群の関係。

2009/05/22

15:00-16:30   Room #128 (Graduate School of Math. Sci. Bldg.)
Prof. Steven Zucker (Johns Hopkins University)
The RBS compactification: a real stratified space in
algebraic geometry

2009/04/27

15:30-18:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Prof. Alessandra Sarti (Universite de Poitier) 15:30-16:30
Automorphism groups of K3 surfaces
[ Abstract ]
I will present recent progress in the study of prime order automorphisms of K3 surfaces.
An automorphism is called (non-) symplectic if the induced
operation on the global nowhere vanishing holomorphic two form
is (non-) trivial. After a short survey on the topic, I will
describe the topological structure of the fixed locus, the
geometry of these K3 surfaces and their moduli spaces.

Prof. Samuel Boissier (Universite de Nice
) 17:00-18:00
The cohomological crepant resolution conjecture


[ Abstract ]
The cohomological crepant resolution conjecture is one
form of Ruan's conjecture concerning the relation between the
geometry of a quotient singularity X/G - where X is a smooth
complex variety and G a finite group of automorphisms - and the
geometry of a crepant resolution of singularities of X/G ; it
generalizes the classical McKay correspondence. Following the
examples of the Hilbert schemes of points on surfaces and the
weighted projective spaces, I will present some of the recents
developments of the subject.

2009/02/19

15:50-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
O. F. Pasarescu (Romanian Academy)
・Linear Systems on Rational Surfaces; Applications (15:50--16: 50)

・Some Applications of Model Theory in Algebraic Geometry (17:00 --18:00)

2008/11/26

16:30-18:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Piotr Pragacz
(Banach Institute)
Diagonal subschemes and vector bundles

2008/11/25

16:30-18:00   Room #118 (Graduate School of Math. Sci. Bldg.)
Xavier Roulleau (東大)
Cotangent maps of surfaces of general type
[ Abstract ]
Surfaces are usualy studied and classified via the properties of the pluricanonical maps. For surfaces of general type whose cotangent sheaf is generated by global sections, we propose to study an other map, called the cotangent map, in order to obtain geometric informations on the surface. In this way, we obtain informations on the ampleness of the cotangent sheaf of such a surface. We will illustate this talk with the example of the Fano surface of lines of cubic threefolds.

2008/11/07

16:30-18:00   Room #118 (Graduate School of Math. Sci. Bldg.)
Misha Verbitsky (ITEP and IPMU)
Hyperkaehler SYZ conjecture and stability
[ Abstract ]
Let L be a nef bundle on a hyperkaehler manifold. A Hyperkaehler SYZ conjecture postulates that L is semi-ample. As shown by Matsushita, this implies existence of holomorphic Lagrangian fibrations on hyperkaehler manifolds. It was conjectured by many
people, most recently by Tschinkel, Hassett, Huybrechts and Sawon. We prove that a sufficiently big power of L is effective, assuming that L admits a semi-positive metric. A multiplier ideal version of this argument would give effectivity of L^N for any nef L. The proof uses stability and Boucksom's divisorial
Zariski decomposition.

2008/10/17

13:00-14:30   Room #128 (Graduate School of Math. Sci. Bldg.)
Yongnam Lee (Sogang U.)
Construction of surfaces of general type with pg=0 via
Q-Gorenstein smoothing

2008/04/21

16:30-18:00   Room #122 (Graduate School of Math. Sci. Bldg.)
高木寛通 (東大数理)
Scorza quartics of trigonal spin curves and their varieties of power sums
[ Abstract ]
Our fundamental result is the construction of new subvarieties in the varieties of power sums for the Scorza quartic of any general pairs of trigonal curves and non-effective theta characteristics. This is a generalization of Mukai's description of smooth prime Fano threefolds of genus twelve as the varieties of power sums for plane quartics. Among other applications, we give an affirmative answer to the conjecture of Dolgachev and Kanev on the existence of the Scorza quartic for any general pairs of curves and non-effective theta characteristics.

2008/03/14

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
David Morrison (UC Santa Barbara)
Understanding singular algebraic varieties via string theory
[ Abstract ]
String theory has helped to formulate two major new insights in the study of singular algebraic varieties. The first -- which also arose from symplectic geometry -- is that families of Kaehler metrics are an important tool in uncovering the structure of singular algebraic varieties. The second, more recent insight -- related to independent work in the representation theory of associative algebras -- is that one's understanding of a singular (affine) algebraic variety is enhanced if one can find a non-commutative ring whose center is the coordinate ring of the variety. We will describe both of these insights, and explain how they are related to string theory.

2008/01/29

10:00-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Dmitry KALEDIN (Steklov研究所, 東大数理)
Homological methods in non-commutative geometry, part 11 (last lecture)
[ Reference URL ]
http://imperium.lenin.ru/~kaledin/math/tokyo/

2008/01/22

10:00-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Dmitry KALEDIN (Steklov研究所, 東大数理)
Homological methods in non-commutative geometry, part 10
[ Reference URL ]
http://imperium.lenin.ru/~kaledin/math/tokyo/

2008/01/15

10:00-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Dmitry KALEDIN (Steklov研究所, 東大数理)
Homological methods in non-commutative geometry, part 9
[ Reference URL ]
http://imperium.lenin.ru/~kaledin/math/tokyo/

2008/01/08

10:00-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Dmitry KALEDIN (Steklov研究所, 東大数理)
Homological methods in non-commutative geometry, part 8

2007/12/11

10:00-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Dmitry KALEDIN (Steklov研究所, 東大数理)
Homological methods in non-commutative geometry, part 7

2007/11/27

16:30-18:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Alexander Kuznetsov (Steklov Inst)
Categorical resolutions of singularities
[ Abstract ]
I will give a definition of a categorical resolution of singularities and explain how such resolutions can be constructed.

2007/11/08

16:30-18:00   Room #118 (Graduate School of Math. Sci. Bldg.)
Alexandru DIMCA (Univ Nice )
New restrictions on the fundamental groups of complex algebraic varieties
[ Abstract ]
My talk will be based on joint work with S. Papadima (Bucarest, Romania) and A. Suciu (Boston, USA). First I will recall the basic facts on characteristic varieties $V_k(M)$ associated to rank one local systems on a complex algebraic variety $M$ which are due to Beauville, Simpson and Arapura. Then I will introduce the resonance varities $R_k(M)$, which may be related to the Isotropic Subspace Theorems by Catanese and Bauer. One of the main new results is that for a class of algebraic varieties (the 1-formal ones), the two types of varieties $V_k(M)$ and $R_k(M)$ are strongly related. Applications to right angle Artin groups, Bestvina-Brady groups and to a conjecture by Kollar will be discussed in the end.

2007/10/30

10:00-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Dmitry KALEDIN (Steklov研究所, 東大数理)
Homological methods in Non-commutative Geometry

2007/10/16

10:00-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Dmitry KALEDIN (Steklov研究所, 東大数理)
Homogical methods in Non-commutative Geometry
[ Abstract ]
Of all the approaches to non-commutative geometry, probably the most promising is the homological one, developed by Keller, Kontsevich, Toen and others, where non-commutative eometry is understood as "geometry of triangulated categories". Examples of "geometric" triangulated categories come from representation theory, symplectic geometry (Fukaya category) and algebraic geometry (the derived category of coherent sheaves on an algebraic variety and
various generalizations). Non-commutative point of view is expected to be helpful even in traditional questions of algebraic geometry such as the termination of flips.

We plan to give an introduction to the subject, with emphasis on homological methods (such as e.g. Hodge theory which, as it turns out, can be mostly formulated in the non-commutative setting).

No knowledge of non-commutative geometry whatsoever is assumed. However, familiarity with basic homological algebra and algebraic geometry will be helpful.

2007/10/10

16:30-17:30   Room #117 (Graduate School of Math. Sci. Bldg.)
James Lewis (University of Alberta)
Abel-Jacobi Maps Associated to Algebraic Cycles, I.

[ Abstract ]
This talk concerns the Bloch cycle class map from the higher Chow groups to Deligne cohomology of a projective algebraic manifold. We provide an explicit formula for this map in terms of polylogarithmic type currents.

2007/10/10

15:00-16:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Dmitry Kaledin (Steklov Institute)
p-adic Hodge theory in the non-commutative setting
[ Abstract ]
We will explain what is the natural replacement of the notion of Hodge structure in the p-adic setting, and how to construct such a structure for non-commutative manifolds (something which at present cannot be done for the usual Hodge structures, but works perfectly well for the p-adic analog).

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