Varieties with torus actions: algebra and combinatorics

本课题研究环面簇的定义方程和高阶合子，即代数环面的等变紧化。一般来说，对于给定的嵌入，很难找到显式方程。然而，人们通常可以确定定义方程的次数以及更高合度的次数，并且该信息揭示了嵌入的几何性质。例如，非线性极小合系的存在与割平面的存在有关。一个环簇嵌入到射影空间对应于一个格多面体，这个项目的目的是进一步理解嵌入的定义方程和合子与多面体的组合性质之间的相互作用。关于环簇的显式结果也可以用于给出不是环簇但允许环簇退化的生成元的次数和更高合度的界。例如，我们将它应用于射影直线上的点的模空间。代数几何是一个与许多其他领域相互作用的数学领域，如数论、拓扑学、数学物理和组合学。代数几何中所研究的空间的最基本构件是消失的多项式方程组。然而，一般而言，所讨论的空间是抽象地给出的，人们通常不知道这些多项式方程是什么。这个项目是关于研究一类特殊的几何空间的这个问题，这些几何空间展示了额外的组合结构，允许比一般情况下更具体的方法。

This project deals with studying the defining equations and higher syzygies of toric varieties, i.e., equivariant compactifications of algebraic tori. In general, it is difficult to find explicit equations for a given embedding. However, often one can determine the degrees of the defining equations as well as the degrees of the higher syzygies, and this information reveals geometric properties of the embedding. For example, the existence of nonlinear minimal syzygies is related to the existence of secant planes. An embedding of a toric variety into projective space corresponds to a lattice polytope, and the goal of this project is to further the understanding of the interplay between the defining equations and syzygies of the embedding and the combinatorial properties of the polytope. Explicit results for toric varieties can also be useful in giving bounds of the degrees of the generators and higher syzygies of varieties that are not toric, but admit a degeneration to a toric variety. For example, we apply this to the moduli space of points on the projective line.Algebraic geometry is an area of mathematics that interacts with many other areas, such as number theory, topology, mathematical physics, and combinatorics. The most basic building blocks of the spaces studied in algebraic geometry are vanishing sets of polynomial equations. However, in general, the space in question is given abstractly, and often one does not know what these polynomial equations are. This project is about studying this question for a special class of geometric spaces that exhibit additional combinatorial structure allowing a more concrete approach than in the general case.