会议名称：Algebraic geometry, number theory, and related topics（中韩双边学术会议）

会议时间：2015.05.20-05.22

韩国8位报告人：Kiryong Chung(KIAS), Seung-Jo Jung(KIAS),Kwang-Seob Kim (KIAS), Sijong Kwak(KAIST)，Jun Ho Lee(KIAS), Kyungyong Lee(Wayne State Univ. and KIAS CMC),Jinhyung Park(KIAS), Yoon Kyung Park (KIAS).

院内8位报告人：付保华、李文威、孙斌勇、田一超、田野、魏达盛、郑维喆、王崧

会议地点：中科院数学院

2015年5月20日上午

1.     Speaker: Baohua Fu

Title: On special birational transformations

Abstract:

A birational transformation f: P^n --> Z, where Z is a nonsingular variety of Picard number 1, is called a special birational transformation of type (a, b) if f is given by a linear system of degree a, its inverse is given by a linear system of degree b and the base locus S \subset P^n off is irreducible and nonsingular. 
I'll report a joint work with Jun-Muk Hwang on the classification of special birational transformations of type (2,1).

2.     Speaker: Jinhyung Park

Title: Syzygies of Cox rings of del Pezzo surfaces

Abstract: 

There has been considerable interest in understanding Cox rings of del Pezzo surfaces. The main problem was the Batyrev-Popov conjecture which describes the relations of generators of those rings, and Testa-Varilly-Alvarado-Velasco and Sturmfels-Xu independently verified this conjecture a few years ago. It is then natural to study syzygies of the relations. After showing basic properties of Cox rings of del Pezzo surfaces, some important invariants such as Hilbert functions, projective dimensions, and Castelnuovo-Mumford regularities are calculated. Based on this calculation, I give an alternative proof of the Batyrev-Popov conjecture and compute the graded betti numbers of Cox rings of some del Pezzo surfaces. This is joint work with Joonyeong Won.

2015年5月20日下午

3.     Speaker: Wen-Wei Li Title: What are zeta integrals?

Abstract:The use of zeta integrals is pervasive in the study of automorphic forms, as exemplified in Tate's famous thesis. Nevertheless, the relevant integrals and test functions are usually crafted case-by-case and require much ingenuity. I will survey some aspects of these integrals, and try to sketch a conjectural unifying framework.

4.     Speaker: Jun Ho Lee

Title: On some orders generated by units

Abstract: 

We are interested in an arithmetic of some orders generated by  a unit(or units). Especially, we focus on the problem of determining a system of fundamental units of those orders. We introduce known results in the case that the rank of the unit group is 1. When the rank of the unit group is 2, we will share our results. This is a joint work with St\'{e}phane R. Louboutin.

5.     Speaker: Dasheng Wei

Title: Strong approximation for a normic variety

Abstract：

For an algebraic variety, strong approximation may imply the existence of integral points of its integral model. In this talk, I will introduce some varieties which satisfy strong approximation. It's a joint work with Ulrich Derenthal.

2015年5月21日上午

6. Speaker: Kiryong Chung

Title: Moduli of sheaves, Fourier-Mukai transform, and partial desingularization

Abstract:

We study birational maps among 1) the moduli space of semistable torsion sheaves of Hilbert polynomial 4m+2 on a smooth quadric surface, 2) the moduli space of semistable torsion sheaves of Hilbert polynomial m^2+3m+2 on P^3, 3) Kontsevich's moduli space of genus zero stable maps of degree 2 to Grassmannian Gr(2,4). A regular birational morphism from 1) to 2) is described in terms of Fourier-Mukai transform. The map from 3) to 2) is Kirwan's partial desingularization. Also we investigate several geometric properties of 1) by using the variation of moduli spaces of stable pairs. This is joint work with Han-Bom Moon (Fordham univ.).

6.     Speaker: Song Wang

Title: Grunwald-Wang

Abstract：

The Grunwald-Wang Theorem, a classical theorem in class field theory, was first formulated and claimed by W. Grunwald in 1933, and finally re-formulated and proved by Shiang-Hao Wang in 1950.

It asserts that given a family of local multiplicative character

$\{ \chi^{v}，v \in S\}$ where $\chi^{v}: F_{v}^{\times} \to \mathbb{C}^{\times}$,

there is a global multiplicative character $\chi$ of $\mathbb{A}_{F}^{\times}$

whose order is minimal, and its local components at given places are given ones.

In this talk, we will review this theorem, and also a proof of effective version,

namely, not only we can prove the existence of such $\chi$, but $N (\chi)$

can be bounded in an effective way.

8．Kwang-Seob Kim

Title: Remarks on quadratic fields with noncyclic ideal class groups. 

Abstract:

Let $n$ be an any integer. It is well known that there are infinitely many imaginary quadratic fields with ideal class group having a subgroup isomorphic to $\Z/n\Z \times \Z/n\Z$. For real quadratic fields, less is known. We will prove that there exist infinitely many real quadratic number fields with ideal class group having a subgroup isomorphic to $\Z_n \times \Z_n$. We will also prove that there exist infinitely many imaginary quadratic number fields with ideal class group having a subgroup isomorphic to $\Z_n \times \Z_n \times \Z_n$. 

2015年5月21日下午

9．Speaker: Weizhe Zheng

Title：Orthogonal and symplectic pure perverse sheaves

Abstract：

It is well known that the odd-degree Betti numbers of projective smooth varieties are even. In this talk, I will discuss parity properties in more general settings, including parity of the odd-weight parts of usual cohomology and intersection cohomology. We deduce these properties from stability properties of orthogonal and symplectic pure perverse sheaves under cohomological operations. This is joint work with Shenghao Sun.

10．Speaker: Kyungyong Lee

Title: Quiver mutations and their applications to algebraic geometry and number theory

Abstract:

A quiver is a directed graph, and its mutation is a certain operation which transforms a quiver to another one. We introduce this notion and discuss some applications.

11．Speaker: Binyong Sun

Title: Conservation relations for local theta correspondence

Abstract:

I will explain the conservation relations for the first occurrence of local theta correspondence, as first conjectured by Kudla and Rallis. This is a joint work with Chen-Bo Zhu of NUS.

2015年5月22日上午

12．Speaker: Seung-Jo Jung

Title: The McKay correspondence and the terminal quotient singularities in dimension three

Abstract:
The McKay correspondence says that for a finite group G acting on a variety M, a certain resolution of the quotient singularity M/G can be realised as a moduli space of G-equivariant objects on M.
For instance, it is well-known that if G is a finite subgroup in SL(2,C), then the minimal resolution of C^2/G is a fine moduli space of a certain G-equivariant sheaves on C^2.
This talk shows that for a 3-fold terminal quotient singularity C^3/G, the economic resolution of the singularity has a moduli interpretation in terms of G-equivariant sheaves on C^3.

13．Speaker: Ye Tian

Title: Explicit Gross-Zagier Formula and arithmetic application

Abstract:

In this talk, we will introduce explicit forms of Gross-Zagier formula and Waldspurger formula and their application to arithmetic problems.

14．Speaker: Sijong Kwak

Title: Graded Betti Numbers and Classification of Algebraic Varieties

Abstract:

In this talk, we introduce the graded Betti numbers of projective coordinate ring associated to algebraic variety. The upper bound of them and nearly extremal cases can be classified or characterized with examples.

15．Speaker：Yichao Tian

Title: Construction of Tate cycles on certain unitary Shimura varieties over finite fields

Abstract:

Let F be a quadratic real field, and E/F be a CM extension. Let p be a prime number inert in F and split in E. Consider  a Shimura variety of PEL type associated to a unitary group over Q of archimedean type  G(U(1,n-1)*U(n-1,1)) , and denote by X its fiber in characteristic p. We will construct n series of algebraic cycles in X such that each of them is parametrized by another unitary Shimura variety of type G(U(0,n)*U(n,0)). We prove that in the generic case, these algebraic cycles give rise to almost all obvious Tate cycles on X. This is a work joint with Liang Xiao and David Helm.

16．Speaker: Yoon Kyung Park 

Title: Periods of modular forms on level p and Jacobi theta functions

Abstract:

We give a closed formula for the expression of sums of period polynomials multiplied by its associated Hecke eigenform over the orthogonal basis of thespace of modular forms on level p for any prime p, by means of sums of productsof Jacobi theta series. This extends the result proved by Zagier's result in 1991.This work is on the progress for more general level and joint with YoungjuChoie.