Geometry of Moduli Spaces of Curves and Surfaces

首席研究员计划研究Deligne， Knudsen和Mumford的稳定曲线及其高维类似物的模空间的几何，即由Kollar， Shepherd-Barron和Alexeev引入的稳定曲面的模空间。第一个目标是揭示代数曲线紧模空间的双几何。主要猜想的灵感来自于物理学中的镜像对称和线性级数的经典代数几何。用高格可约曲线的紧化雅可比矩阵描述了稳定有理曲线模空间的有效因子锥和（部分）双空间收缩的变化。第二个目标是发展明确描述稳定正则极化曲面模空间的新方法，并将这些方法应用于几种经典情况。代数几何研究代数变异：由多项式方程组定义的形状。代数变量具有离散特征，允许对其种类进行分类：有理曲线，Del Pezzo曲面，Abelian变量，Calabi-Yau变量，一般类型的变量等。每种类型的变种都依赖于某些连续参数（称为模），所有参数的集合具有丰富的所谓模空间结构。一个特别感兴趣的是紧模空间，它参数化了允许轻度退化的变量。例如，平面上的双曲线xy=C，当C趋于0时，可以退化为两条直线xy=0的并集。主要研究这些紧模空间及其在代数几何中的相关问题。这一具有数百年历史的纯数学概念与物理学有着丰富的关系，并且将其应用于计算代数几何将导致在代数统计和数学生物学中有用的新算法。

The principal investigator plans to study geometry of moduli spaces of stable curves of Deligne, Knudsen, and Mumford and its higher dimensional analogues, namely the moduli spaces of stable surfaces introduced by Kollar, Shepherd-Barron, and Alexeev. The first goal is to uncover birational geometry of the compact moduli space of algebraic curves. The main conjecture is inspired by Mirror Symmetry from physics and classical algebraic geometry of linear series. It describes the cone of effective divisors and (some part of) the variation of birational contractions of the moduli space of stable rational curves in terms of compactified Jacobians of reducible curves of high genus. The second goal is to develop new methods of explicitly describing the moduli spaces of stable canonically polarized surfaces and to apply these methods in several classical situations.Algebraic geometry studies algebraic varieties: shapes defined by systems of polynomial equations. Algebraic varieties have discrete characteristics that allow to classify their species: rational curves, Del Pezzo surfaces, Abelian varieties, Calabi-Yau varieties, varieties of general type, etc. Varieties of each type depend on certain continuous parameters (called moduli) and the set of all parameters has a rich structure of the so-called moduli space. One is particularly interested in compact moduli spaces that parametrize varieties with allowed mild degenerations. For example, a hyperbola xy=C on the plane can degenerate to the union of two lines xy=0 when C goes to 0. The principal investigator will study these compact moduli spaces and related problems in algebraic geometry. This centuries-old concept of pure mathematics has rich relationship with physics, and applications to computational algebraic geometry will lead to new algorithms useful in algebraic statistics and mathematical biology.