代数幾何学セミナー

過去の記録 ~02/26次回の予定今後の予定 02/27~

開催情報 金曜日 13:30~15:00 数理科学研究科棟(駒場) ハイブリッド開催/117号室
担当者 權業 善範、中村 勇哉、田中 公

過去の記録

2014年10月27日(月)

14:50-16:20   数理科学研究科棟(駒場) 122号室
いつもと開始時間が異なります。
Meng Chen 氏 (Fudan University)
On projective varieties with very large canonical volume (ENGLISH)
[ 講演概要 ]
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年07月07日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
谷本翔 氏 (Rice University)
Balanced line bundles (JAPANESE)
[ 講演概要 ]
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年06月30日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
三内顕義 氏 (東京大学数理科学研究科)
Invariant subrings of the Cox rings of K3surfaces by automorphism groups (JAPANESE)
[ 講演概要 ]
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年06月02日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
中村勇哉 氏 (東京大学数理科学研究科)
On base point free theorem for log canonical three folds over the algebraic closure of a finite field (JAPANESE)
[ 講演概要 ]
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年05月12日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Andrés Daniel Duarte 氏 (Institut de Mathématiques de Toulouse)
Higher Nash blowup on normal toric varieties and a higher order version of Nobile's theorem (ENGLISH)
[ 講演概要 ]
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年04月28日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Alexandru Dimca 氏 (Institut Universitaire de France )
Syzygies of jacobian ideals and Torelli properties (ENGLISH)
[ 講演概要 ]
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.

2014年02月12日(水)

14:00-17:30   数理科学研究科棟(駒場) 122号室
いつもと曜日・時間が異なります
松村 慎一 氏 (鹿児島大学) 14:00-15:30
An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities (ENGLISH)
[ 講演概要 ]
In this talk, I give an injectivity theorem with multiplier ideal sheaves of singular metrics.
This theorem is a powerful generalization of various injectivity and vanishing theorems.
The proof is based on a combination of the theory of harmonic integrals and the L^2-method for the \\dbar-equation.
To treat transcendental singularities, after regularizing a given singular metric, we study the asymptotic behavior of the harmonic forms with respect to a family of the regularized metrics.
Moreover we obtain L^2-estimates of solutions of the \\dbar-equation, by using the \\check{C}ech complex.
As an application, we obtain a Nadel type vanishing theorem.
Junyan Cao 氏 (KIAS) 16:00-17:30
Ohsawa-Takegoshi extension theorem for K\\"ahler manifolds (ENGLISH)
[ 講演概要 ]
In this talk, we first prove a version of the Ohsawa-Takegoshi
extension theorem valid for on arbitrary K\\"ahler manifolds, and for
holomorphic line bundles equipped with possibly singular metrics. As an
application, we generalise Berndtsson and Paun 's result about the
pseudo-effectivity of the relative canonical bundles to arbitrary
compact K\\"ahler families.

2014年02月03日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
藤田 健人 氏 (京都大学数理解析研究所)
Classification of log del Pezzo surfaces of index three (JAPANESE)
[ 講演概要 ]
Log del Pezzo surfaces constitute an interesting class of rational surfaces and naturally appear in the minimal model program. I will describe an algorithm to classify all the log del Pezzo surfaces of fixed (Q-Gorenstein) index $a$. Especially, I will focus on the case that $a$ is equal to three. This is joint work with Kazunori Yasutake.

2014年01月22日(水)

15:00-16:30   数理科学研究科棟(駒場) 122号室
いつもと曜日・時間が異なります
Thomas Ducat 氏 (University of Warwick)
Divisorial Extractions from Singular Curves in Smooth 3-Folds (ENGLISH)
[ 講演概要 ]
Consider a singular curve C contained in a smooth 3-fold X.
Assuming the existence of a Du Val general elephant S containing C,
I give a normal form for the equations of C in X and an outline of how to
construct a divisorial extraction from this curve. If the general S is
Du Val of type D_{2k}, E_6 or E_7 then I can give some explicit
conditions for the existence of a terminal extraction. A treatment of
the D_{2k+1} case should be possible by similar means.

2014年01月20日(月)

15:30-17:00   数理科学研究科棟(駒場) 126号室
いつもと部屋が異なります
佐野 太郎 氏 (University of Warwick)
Deforming elephants of Q-Fano 3-folds (ENGLISH)
[ 講演概要 ]
Shokurov and Reid proved that a Fano 3-fold with canonical
Gorenstein singularities has a Du Val elephant, that is,
a member of the anticanonical linear system with only Du Val singularities.
The classification of Fano 3-folds is based on this fact.
However, for a Fano 3-fold with non-Gorenstein terminal singularities,
the anticanonical system does not contain such a member in general.
Alt{\\i}nok--Brown--Reid conjectured that, if the anticanonical system is non-empty,
a Q-Fano 3-fold can be deformed to that with a Du Val elephant.
In this talk, I will explain how to deform an elephant with isolated
singularities to a Du Val elephant.

2013年12月09日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
岡田 拓三 氏 (佐賀大学)
On birationally tririgid Q-Fano threefolds (JAPANESE)
[ 講演概要 ]
I will talk about birational geometry of Q-Fano threefolds. A Mori
fiber space birational to a given Q-Fano threefold is called a birational Mori fiber structure of the threefold. The existence of Q-Fano threefolds with a unique birational Mori fiber structure (resp. with two birational Mori fiber structures) is known. In this talk I will give an example of Q-Fano threefolds with three birational Mori fiber structures and also discuss about the behavior of birational Mori fiber structures in a family.

2013年11月25日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
小池 貴之 氏 (東大数理)
Minimal singular metrics of some line bundles with infinitely generated section rings (JAPANESE)
[ 講演概要 ]
We consider Hermitian metrics of pseudo-effective line bundles on smooth
projective varieties defined over $\\mathbb{C}$.
Especially we are interested in (possibly singular) Hermitian metrics
with semi-positive curvatures when the section rings are not finitely generated.
We study where and how minimal singular metrics, special Hermitian
metrics with semi-positive curvatures, diverges in the following two situations;
a line bundle admitting no Zariski decomposition even after any
modifications (Nakayama example)
and a nef line bundle $L$ on $X$ satisfying $D \\subset |mL|$ and $|mL-D|
= \\emptyset$ for some divisor $D \\subset X$ and for all $m \\geq 1$ (
Zariski example).

2013年11月18日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
この講演はGCOEレクチャーズとの共同セミナーになります
Florin Ambro 氏 (IMAR)
An injectivity theorem (ENGLISH)
[ 講演概要 ]
I will discuss a generalization of the injectivity theorem of Esnault-Viehweg, and an
application to the problem of lifting sections from the non-log canonical locus of a log variety.

2013年11月11日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Sung Rak Choi 氏 (POSTECH)
Geography via the base loci (ENGLISH)
[ 講演概要 ]
The geography of log model refers to the decomposition of the set of effective adjoint divisors into the cells defined by the resulting models that are obtained by the log minimal model program.
We will describe the geography in terms of the asymptotic base loci and Zariski decompositions of divisors.
As an application, we give a partial answer to a question of B. Totaro concerning the structure of partially ample cones.

2013年10月28日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
江 辰 氏 (東京大学数理科学研究科)
Weak Borisov-Alexeev-Borisov conjecture for 3-fold Mori Fiber spaces (ENGLISH)
[ 講演概要 ]
We investigate $\\epsilon$-klt log Fano 3-folds with some Mori fiber space structure, more precisely, with a del Pezzo fibration structure, or a conic bundle structure over projective plane. We give a bound for the log anti-canonical volume of such pair. The method is constructing non-klt centers and using connectedness lemma. This result is related to birational boundedness of log Fano varieties.

2013年07月22日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Stavros Papadakis 氏 (RIMS)
Equivariant degenerations of spherical modules (ENGLISH)
[ 講演概要 ]
Given a reductive algebraic group G and an invariant
Hilbert function h, Alexeev and Brion have defined
a moduli scheme M which parametrizes affine G-schemes X
with the property that the coordinate ring of X decomposes,
as G-module, according to the function h. The talk will
be about joint work with Bart Van Steirteghem (New York)
which studies the moduli scheme M under some additional
assumptions.

2013年04月22日(月)

16:30-18:00   数理科学研究科棟(駒場) 118号室
Professor Igor Reider 氏 (Universite d'Angers / RIMS)
Kodaira-Spencer classes, geometry of surfaces of general type and Torelli
theorem (ENGLISH)
[ 講演概要 ]
In this talk I will explain a geometric interpretation of Kodaira-Spencer classes and apply
it to the study of the differential of the period map of weight 2 Hodge structures for surfaces
of general type.
My approach is based on interpreting Kodaira-Spencer classes as higher rank bundles and
then studing their stability. This naturally leads to two parts:
1) unstable case
2) stable case.
I will give a geometric characterization of the first case and show how to relate the second
case to a special family of vector bundles giving rise to a family of rational curves. This family
of rational curves is used to recover the surface in question.

2013年01月15日(火)

15:30-17:00   数理科学研究科棟(駒場) 128号室
いつもと曜日・場所が異なります
Jungkai Alfred Chen 氏 (National Taiwan University)
Three Dimensional Birational Geoemtry--updates and problems (ENGLISH)
[ 講演概要 ]
In this talk I will talk about some recent results on
biratioanl classification and biratioanl geoemtry of threefolds.

Given a threefold of general type, we improved our previous result by
showing that $Vol \\ge 1/1680$ and $|mK_X|$ is biratioanl for $m \\ge
61$.
Compare with the worst known example that $X_{46} \\subset
\\mathbb{P}(4,5,6,7,23)$, one also knows that there are only finiteley
many singularities type
for threefolds of general type with $1/1680 \\le Vol \\le 1/420$. It is
then intereting to study threefolds of general type with given basket
of singularities and with given fiber structure.
Concerning threefolds with intermediate Kodaira dimension, we
considered the effective Iitaka fibration. For this purpose, it is
interesting to study threefolds with $\\kappa=1$ with given basket of
singularities and abelian fibration.

For explicit birational geoemtry, we will show our result that each
biratioanl map in minimal model program can be factored into a
sequence of following maps (or its inverse)
1. a divisorial contraction to a point of index r with discrepancy 1/r.
2. a blowup along a smooth curve
3. a flop

2012年12月13日(木)

10:40-12:10   数理科学研究科棟(駒場) 118号室
いつもと曜日・時間・場所が異なります
Jean-Paul Brasselet 氏 (CNRS (Luminy))
The asymptotic variety of polynomial maps (ENGLISH)
[ 講演概要 ]
The asymptotic variety, or set of non-properness has been intensively studied by Zbigniew Jelonek. In a recent paper, Anna and Guillaume Valette associate to a polynomial map $F: {\\mathbb C}^n \\to {\\mathbb C}^n$ a singular variety $N_F$ and relate properness property of $F$ to the vanishing of some intersection homology groups of $N_F$. I will explain how stratifications of the asymptotic variety of $F$ play an important role in the story and how recently, one of my students, Nguyen Thi Bich Thuy, found a nice way to exhibit such a suitable stratification.

2012年12月10日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
川谷 康太郎 氏 (名古屋大学多元数理科学研究科)
A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)
[ 講演概要 ]
We introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank 1. Furthermore we demonstrate how this hyperbolic metric is helpful for us by discussing two or three topics.

2012年11月26日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
桂 利行 氏 (法政大学理工学部)
超特殊K3曲面上の有理曲線の配置について (JAPANESE)
[ 講演概要 ]
正標数の代数的閉体$k$上の超特異K3曲面のArtin不変量が1のとき超特殊K3曲面という。標数が3以上であれば、このようなK3曲面は、2つの超特異楕円曲線の直積であるアーべル曲面からつくられるKummer曲面になることが知られている。この講演では$S$上の有理曲線の配置をアーベル曲面の因子の構造を用いて考察し、標数が2ならば$(21)_5$-symmetric configurationが存在すること、また標数3ならば$(16)_{10}$-symmetric configurationと$(280_{4}, 112_{10})$-configurationが存在することを示す。また、後者は、$p^{a} + 1$次のFermat hypersurfaceのline configurationや、N\\'eron-S\\'everi群${\\rm NS}(S)$がLeech latticeを用いて捉えられることと関係することを述べる。

2012年11月19日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
戸田 幸伸 氏 (IPMU)
Stability conditions and birational geometry (JAPANESE)
[ 講演概要 ]
I propose a conjecture which claims that MMP for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, I will construct the perverse t-structure associated to the extremal contraction, and construct a candidate of the desired stability condition as a double tilting of the perverse heart.

2012年11月12日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
安武 和範 氏 (九州大学大学院数理学研究院)
On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)
[ 講演概要 ]
By using results about extremal contractions on smooth fourfolds, we give a classification of fano fourfolds whose the second exterior power of tangent bundles are numerically effective.

2012年11月05日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
馬 昭平 氏 (名古屋大学多元数理科学研究科)
トリゴナル曲線のモジュライの有理性 (JAPANESE)
[ 講演概要 ]
射影直線を3対1で被覆するような曲線はトリゴナル曲線と呼ばれ、超楕円曲線に次いで特殊な曲線である。Shepherd-Barron氏は種数が4で割って2余る場合にトリゴナル曲線のモジュライ空間が射影空間と双有理同型であることを証明した。本講演ではその結果を全ての種数に拡張する。

2012年10月29日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
藤田 健人 氏 (京都大学数理解析研究所)
The Mukai conjecture for log Fano manifolds (JAPANESE)
[ 講演概要 ]
The concept of log Fano manifolds is one of the most natural generalization of the concept of Fano manifolds. We will give some structure theorems of log Fano manifolds. For example, we will show that the Mukai conjecture for Fano manifolds implies the `log Mukai conjecture' for log Fano manifolds.

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