Remarks on Theta-stratifications and derived categories

Theta-stratifications, Theta-reductive stacks, and applications

These are expanded notes on a lecture of the same title at the 2015 AMS summer institute in algebraic geometry. We give an introduction and overview of the "beyond geometric invariant theory" program for analyzing moduli problems in algebraic geometry. We discuss methods for analyzing stability in general moduli problems, focusing on the moduli of coherent sheaves on a smooth projective scheme as a toy example. We describe several applications: a general structure theorem for the derived category of coherent sheaves on an algebraic stack; some results on the topology of moduli stacks; and a "v…

超椭圆曲线θ除子的对数消解

A log resolution for the theta divisor of a hyperelliptic curve

该研究针对光滑超椭圆曲线θ除子的奇点消解问题，提出通过迭代吹张Brill-Noether子簇得到自然显式的嵌入消解，并证明超椭圆雅可比簇的Brill-Noether分层为Whitney分层，为代数几何中相关奇点研究提供了新的显式构造结果。

The degree of the Gauss map of the theta divisor

We study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. We use this to obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus, and apply this bound in examples, and to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppav's by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension.

Remarks on Ekedahl-Oort stratifications

This short paper is a continuation of the author's Ph.D thesis, where Ekedahl-Oort strata are defined and studied for Shimura varieties of Hodge type. The main results here are as follows. 1. The Ekedahl-Oort stratification is independent of the choices of symplectic embeddings. 2. Under certain reasonable assumptions, there is certain functoriality for Ekedahl-Oort stratifications with respect to morphisms of Shimura varieties.

Theta 函数与阿贝尔簇上的绝热曲率

Theta functions and adiabatic curvature on an Abelian variety

本文针对阿贝尔簇上丰沛线丛族，利用 theta 函数与微分几何方法，显式计算了直接像丛的曲率，并探讨了其代数几何性质。

How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties

Let $\mathcal D_{d,k}$ denote the discriminant variety of degree $d$ polynomials in one variable with at least one of its roots being of multiplicity $\geq k$. We prove that the tangent cones to $\mathcal D_{d,k}$ span $\mathcal D_{d,k-1}$ thus, revealing an extreme ruled nature of these varieties. The combinatorics of the web of affine tangent spaces to $\mathcal D_{d,k}$ in $\mathcal D_{d,k-1}$ is directly linked to the root multiplicities of the relevant polynomials. In fact, solving a polynomial equation $P(z) = 0$ turns out to be equivalent to finding hyperplanes through a given point $P(…