Varieties with torus actions: algebra and combinatorics

本项目涉及研究复曲面品种的定义方程和更高合点，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，通常人们可以确定定义方程的次数以及更高合点的次数，并且这些信息揭示了嵌入的几何性质。例如，非线性极小合合的存在性与割线平面的存在性有关。一个环面簇到射影空间的嵌入对应于一个格多面体，这个项目的目标是进一步理解定义方程和嵌入的合偶之间的相互作用和多面体的组合性质。明确的结果复曲面品种也可以是有用的，在给界的程度的发电机和更高的syzygies的品种，不是复曲面，但承认退化的复曲面品种。代数几何是一个与数论、拓扑学、数学物理和组合学等许多领域相互作用的数学领域。代数几何中研究的空间的最基本的构建块是多项式方程的消失集。然而，一般来说，所讨论的空间是抽象的，人们往往不知道这些多项式方程是什么。这个项目是关于研究这个问题的一类特殊的几何空间，展示额外的组合结构，允许一个更具体的方法比一般情况下。

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.