Geometry of Compact Moduli Spaces

主要研究者计划研究Deligne，Knudsen和Mumford的稳定曲线的模空间的几何及其高维类似物：由Kollar，Shepherd-Barron和Alexeev引入的稳定对的模空间。最深刻的结果预计模的德尔佩佐表面和某些类的表面的一般类型，模的稳定合理的曲线，并为他们的高维类似物：紧凑的模空间的超平面安排。新的方法是基于森理论和热带代数几何之间的相互作用，它分配给任何代数品种的组合性质的对象，基本上是一个多面体。这个多面体（一个热带变种）令人惊讶地编码了很多关于这个变种的几何信息，最重要的是关于它的紧化。热带簇最初是在计算代数几何学家中引入的，他们研究如何从方程中读取代数簇的不变量。事实证明，所提出的方法也可以向后使用，以推进计算代数几何。代数几何研究由多项式方程组定义的代数簇的形状。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。人们对紧模空间特别感兴趣，它可以参数化允许轻度退化的簇。例如，平面上的双曲线xy=C可以退化为两条直线xy=0的并，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: moduli spaces of stable pairs introduced by Kollar, Shepherd-Barron, and Alexeev. The deepest results are expected for moduli of Del Pezzo surfaces and certain classes of surfaces of general type, for moduli of stable rational curves, and for their higher-dimensional analogues: compact moduli spaces of hyperplane arrangements. New methods are based on the interaction between Mori theory and tropical algebraic geometry, which assigns to any algebraic variety an object of combinatorial nature, basically a polytope. This polytope (a tropical variety) surprisingly encodes a lot of geometric information about the variety and most importantly about its compactifications. Tropical varieties were originally introduced in the community of computational algebraic geometers who study how to read invariants of algebraic varieties from their equations.It turns out that proposed methods can also be used backwards to advance computational algebraic geometry. In particular, they can be used to find implicit equations of algebraic varieties given parametrically.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.