Moduli spaces of curves and surfaces

PI将研究代数几何中的几个基本问题，重点是稳定曲线和稳定曲面的紧致模空间（由Kollar，Shepherd-Barron和Alexeev介绍）。目标之一是在这些模空间上引入和描述各种有效因子，并研究它们如何影响它们的双有理几何。提出了构造一般类型稳定曲面的新方法，并详细研究了单连通伪del Pezzo曲面的模空间。代数几何研究代数簇：由多项式方程组定义的形状。品种分类离散（拓扑）和连续（几何）参数，称为模量。模空间的研究使我们能够理解给定形状上所有可能的几何结构。拟议的研究计划与其他学科有几个联系点，例如Landau-Ginzburg模型的引入使各种一般类型的模空间对物理学家有吸引力。

The PI will work on several fundamental problems in algebraic geometry with a focus on compact moduli spaces of stable curves and stable surfaces (introduced by Kollar, Shepherd-Barron, and Alexeev). One of the goals is to introduce and describe various effective divisors on these moduli spaces and study how they affect their birational geometry. New methods of constructing stable surfaces of general type are proposed, with the moduli space of simply connected fake del Pezzo surfaces studied in detail.Algebraic geometry studies algebraic varieties: shapes defined by systems of polynomial equations. Varieties are classified by discrete (topological) and continuous (geometric) parameters, called moduli. The study of the moduli space allows us to understand all possible geometric structures on a given shape. The proposed research program has several points of contact with other disciplines, for example introduction of Landau-Ginzburg models made moduli spaces of varieties of general type attractive to physicists.