2015年度
後期
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日時: 平成28年02月24日（水）10:30--
講演者: 大川新之介 氏（大阪大学）
タイトル: Compact moduli of marked noncommutative del Pezzo surfaces via quivers
アブストラクト: I will introduce certain GIT construction via quivers of compactified moduli
spaces of marked noncommutative del Pezzo surfaces.
These moduli spaces naturally contain moduli of marked del Pezzo surfaces in an
appropriate sense. For projective plane, quadric surface, and del Pezzo surfaces
of degree 3, 2, 1, our moduli spaces are projective toric varieties of
dimension 2, 3, 8, 9, 10, respectively.
I will also discuss relations with deformation theory of abelian categories,
blow-up of noncommutative projective planes,
and three-block exceptional collections due to Karpov and Nogin.
This talk is based on joint works with Tarig Abdelgadir and Kazushi Ueda.
- 平成28年1月8日（金）15:00--
講演者: Alex Degtyarev 氏(Bilkent大学, JSPS海外特別研究員)
タイトル: Real Enriques surfaces
アブストラクト:I will start with a brief introduction to topology of real algebraic
surfaces (or even curves),
stating the basic problems and outlining the general restrictions
on the topology of the real part of a surface imposed by that of its
complexification.
This part of the talk is mainly of a topological nature.
Then, I will state our main classification theorem (in short, real Enriques
surfaces are quasi-simple, that is to say that the deformation class
of such a surface is determined by the topology of its real structure).
By definition, an Enriques surface is the quotient of a K3-surface by a
fixed point free anti-symplectic involution. Thus, in
principle, one should be able to lift the real structure to the covering
K3-surface and study the problem by means of the well-established theory of
K3-surfaces and Nikulin's theory of discriminant forms.
We do use this approach in the so-called elliptic case,
when all components of the real part have positive Euler characteristic. In
the other cases, the resulting arithmetical problem does not seem very
transparent, and we use an alternative approach, the so-called
Donaldson trick (actually due to Hitchin): by rebuilding the complex
structure on the underlying
4-manifold, we reduce the problem to a question
about a real anti-bicanonical curve on a real rational surface
(a simple example being a sextic curve in the plane); the
latter is treated by means of very classical algebraic geometry.
This is a joint work with Ilia Itenberg and Viatcheslav Kharlamov.
- 平成27年12月4日（金）15:00--
講演者: Viacheslav Nikulin 氏(University of Liverpool)
タイトル:Classification of degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups
アブストラクト:We classify degenerations of codimension one of Kahlerian K3
surfaces with finite symplectic automorphism groups.
By classification we understand an enumeration
of connected components of moduli spaces of the degenerations.
See our preprints arXiv:1403.6061 and 1504.00326 for some details.
- 平成27年11月27日（金）15:00--
講演者: 平之内俊郎 氏 (広島大学)
タイトル: 有限体上の多様体における Hermite-Minkowski 型有限性定理について
アブストラクト: 有限体上の多様体のエタール被覆が次数と分岐を制限すれば
有限個しかないことを示す。これは古典的な Hermite-Minkowski の定理の
関数体版の高次元化になっている。
- 平成27年10月9日（金）15:00--
講演者: Alex Degtyarev 氏 (Bilkent大学, JSPS海外特別研究員)
タイトル: LINES ON SMOOTH QUARTICS
アブストラクト: In 1943, B. Segre proved that a smooth quartic surface in the
complex projective space cannot contain more than 64 lines. (The champion,
so-called Schur's quartic, has been known since 1882.) Even though a gap was
discovered in Segre's proof (Rams, Schütt, 2015), the claim is still correct;
moreover, it holds over any of characteristic other than 2 or 3.
(In characteristic 3, the right bound seems to be 112.) At the same time,
it was conjectured that not any number between 0 and 64 can occur as the
number of lines in a quartic.
We tried to attack the problem using the theory of $K3$-surfaces and arithmetic
of lattices. This relatively simple reduction has lead us to an extremely difficult
arithmetical problem. Nevertheless, the approach turned out quite fruitful: for the
moment, we have a complete classification of smooth quartics containing more than 52 lines.
As an immediate consequence of this classification, we have the following:
-- an alternative proof of Segre's bound 64;
-- Shur's quartic is the only one with 64 lines;
-- a real quartic may contain at most 56 real lines;
-- a real quartic with 56 real lines is also unique;
-- the number of lines takes values {0,...,52,54,56,60,64}.
Conjecturally, we have a complete list of all quartics with more than 48 lines;
there are about two dozens of species, most projectively rigid.
I will discuss methods used in the proof and a few problems that are still open,
e.g., the minimal fields of definition, triangle-free configurations, lines in
singular quartics, etc. This subject is a joint work in progress with Ilia Itenberg
and Sinan Sertöz.
前期
- 平成27年07月31日（金）15:00--
講演者: Nguyen Chanh Tu 氏 (Danang University of Technology)
タイトル: Recent results on regularity index of $n+3$ fat points in the n-dimensional projective space
アブストラクト: A set of fat points is called {\it almost equimultiple}
if the multiplicities of the points are equal to $k$ or $k-1$ for
a given integer $k\ge 2$. In the talk, we use the algebraic method
to prove a well-known conjecture about the upper bound of the regularity index
for any set of $n+3$ almost equimultiple, non-degenerate fat points in $\p^n$.
We also talk about the recent results on the problem.
- 平成27年07月24日（金）月曜日の振替授業
- 平成27年07月17日（金）広島仙台整数論集会
- 平成27年07月10日（金）編入学試験のためお休み
- 平成27年07月3日（金）15:00--
講演者: 平野雄貴 氏(首都大学東京)
タイトル: Equivalences of derived factorization categories of gauged Landau-Ginzburg models
アブストラクト: For a given Fourier-Mukai equivalence of bounded derived categories of coherent sheaves
on smooth quasi-projective varieties, we construct Fourier-Mukai equivalences of derived factorization categories
of gauged Landau-Ginzburg (LG) models. This result is an equivariant version of the result of Baranovsky and Pecharich.
As an application, we obtain some equivalences of derived factorization categories of K-equivalent gauged LG models,
which gives a partial answer to Segal’s conjecture. As another application, we prove that if the kernel of the Fourier-Mukai equivalence
of derived categories is linearizable with respect to a reductive affine algebraic group action, then the derived categories of
equivariant coherent sheaves on the varieties are equivalent. This result is shown by Ploog for finite groups case.
- 平成27年06月05日（金）15:00 --
講演者: 吉永正彦 氏 (北海道大学)
タイトル: 超平面配置とオイラー多項式
アブストラクト: オイラー多項式は元々オイラーがゼータ関数の特殊値を記述するために
導入したものであるが、様々な組合せ論的性質が研究されている。
最近明かになった、（A型の） Linial 配置と呼ばれる配置の特性多項式と
オイラー多項式の関係を報告したい。時間が許せば、ルート系に関する
Postnikov と Stanley による『リーマン予想』（Linial 配置の特性多項式の
零点が実部一定の直線上に並ぶ）の部分的結果を紹介したい。
（参考文献 arXiv:1501.04955）
- 平成27年05月26日（火）15:00 -- (いつもと曜日・場所が違います)
場所: 広島大学理学部C棟C816号室
講演者: Anthony Giaquinto 氏 (Loyola University)
タイトル: Some Properties of Frobenius Lie Algebras
アブストラクト: A Lie algebra is Frobenius if there exists a linear functional F such that
the bilinear form F([x, y]) is non-degenerate. The relation between Frobenius Lie algebras
and the classical Yang-Baxter equation (CYBE) is well-studied. In this talk I will show
how to explicitly construct solutions to the CYBE from certain graphs associated to the linear functional F.
I will also report on the principal element, the Ooms Spectrum, and other interesting properties and
conjectures about Frobenius Lie algebras.
- 平成27年05月22日（金）15:00 --
講演者: James Lewis 氏 (University of Alberta)
タイトル: An Archimedean Height Pairing on the Equivalence Relation
Defining Bloch's Higher Chow Groups
アブストラクト: The existence of a height pairing on the equivalence relation
defining Bloch's higher Chow groups is a surprising consequence of
some recent joint work by myself and Xi Chen. I will begin with an elementary
motivating example situation, define the higher Chow groups in a user friendly way, and
explain how this pairing comes about.
- 平成27年05月01日（金）15:00 --
講演者: Daniel Allcock 氏 (University of Texas)
タイトル: Hyperbolic reflection groups and automorphisms of Enriques surfaces
- 平成27年04月24日（金）15:00 --
講演者: Igor Dolgachev 氏(University of Michigan, RIMS)
タイトル: Pencils of quadrics in characteristic 2
アブストラクト: I will describe normal forms of a pencil of quadratic forms
on an odd-dimensional linear space over an algebraically closed field
of characteristic 2 satisfying the condition that the base locus
of the corresponding pencil of quadric hypersurfaces in the associated projective space
is nonsingular. This applies to a possible structure of the automorphism group of the base locus.
We also give an arithmetical application proving the unirationality of a quartic
del Pezzo surface over a perfect field of characteristic 2.
This is a report on a joint work with Alex Duncan.
- 平成27年04月03日（金）13:30 -- 15:00 (いつもと開始時間が違います)
講演者: JongHae Keum 氏 (KIAS)
タイトル: Automorphisms of K3 surfaces
アブストラクト: I will report recent progress on this topic, mostly in positive characteristic.