Geometry of Moduli Spaces of Curves and Surfaces

主要研究者计划研究Deligne，Knudsen和Mumford稳定曲线的模空间的几何及其高维类似物，即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的紧模空间的双有理几何。主要猜想是受物理学和线性级数的经典代数几何中镜像对称性的启发。它用高亏格可约曲线的紧化Jacobian描述了稳定有理曲线模空间的有效因子锥和（部分）双有理压缩的变化.第二个目标是开发新的方法明确描述的模空间的稳定规范极化表面和应用这些方法在几个经典的situations. Algebrahem几何研究代数品种：形状定义的系统多项式方程。代数变种具有离散的特征，可以将它们的种类分类：有理曲线，Del Pezzo曲面，Abel变种，Calabi-Yau变种，一般类型的变种等。一个是特别感兴趣的紧模空间参数化品种允许轻度退化。例如，平面上的双曲线xy=C可以退化为两条直线xy=0的并集，当C变为0时。主要研究者将研究这些紧凑的模空间和代数几何中的相关问题。这个有着几百年历史的纯数学概念与物理学有着丰富的联系，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.