Algebraic Geometry Seminar

Seminar information archive ~11/28Next seminarFuture seminars 11/29~

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


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


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


16:40-18:10   Room #126 (Graduate School of Math. Sci. Bldg.)
Xavier Roulleau (The University of Tokyo)
Genus 2 curve configurations on Fano surfaces (ENGLISH)


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


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


16:40-18:10   Room #126 (Graduate School of Math. Sci. Bldg.)
Yuji Odaka (Research Institute for Mathematical Sciences)
On the GIT stability of Polarized Varieties (JAPANESE)
[ Abstract ]
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.

(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.)


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


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


16:40-18:10   Room #126 (Graduate School of Math. Sci. Bldg.)
松村 慎一 (東大数理)
[ 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.
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


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


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


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


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


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


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


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


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


16:40-18:10   Room #126 (Graduate School of Math. Sci. Bldg.)
渡辺 究 (早稲田大学基幹理工学研究科)
[ Abstract ]


16:40-18:10   Room #126 (Graduate School of Math. Sci. Bldg.)
伊藤 敦 (東大数理)
[ Abstract ]


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Matthias Schuett (Leibniz University Hannover)
Arithmetic of K3 surfaces
[ Abstract ]
This talk aims to review recent developments in the arithmetic of K3 surfaces, with emphasis on singular K3 surfaces.
We will consider in particular modularity, Galois action on Neron-Severi groups and behaviour in families.


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
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


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)
[ 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.


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
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。


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_+.


16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
馬 昭平氏 (東大数理)

[ Abstract ]

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