Combinatorial Commutative Algebra of Cox Rings and the Hilbert Scheme of Points

本课题研究交换代数、组合学和代数几何之间的两个共同点：代数族的Cox环和点的Hilbert格式。代数簇$X$的Cox环是一个多分次环，它的git商是通过有理映射得到的$X$像的坐标环。对于许多有趣的变种，即所谓的Mori Dream空间，Cox环是有限生成的k-代数。该项目旨在开发工具，在显式坐标和定义方程方面有效地实现某些Mori Dream空间的Cox环。这样的描述是通过Koszul过滤、通过Groebner基的组合或通过所提出的新方法来获得的。对某些种类的Cox环的结构提出了具体的猜想。该项目还研究了仿射空间中点的Hilbert格式，特别是描述根分量(根理想集的闭包)和单项理想附近的奇点的问题。代数簇是由多个变量多项式函数的集合同时消失所定义的空间。研究代数族几何的一种非常有用的经典方法是考虑这种代数族所允许的各种自然坐标系。代数族的Cox环是同时研究所有这些坐标系的一种方法。这个项目的第一个目标是给出一些代数簇的Cox环的描述。这个建议的第二个目标是研究点的Hilbert格式：d维空间中n个点的所有可能构形的空间。这个集合的几何结构相当复杂，该项目的目的是描述n个点相互碰撞的所有可能方式，以及这些碰撞附近的希尔伯特方案的几何结构。

This project studies two objects in the common ground between commutative algebra, combinatorics and algebraic geometry: the Cox rings of algebraic varieties and the Hilbert schemes of points. The Cox ring of an algebraic variety $X$ is a multigraded ring whose GIT quotients are the coordinate rings of images of $X$ via rational maps. For many interesting varieties, the so called Mori Dream spaces, the Cox ring is a finitely generated k-algebra. The project aims to develop tools to realize Cox rings of some Mori Dream spaces effectively, in terms of explicit coordinates and defining equations. Such descriptions are obtained via Koszul filtrations, via the combinatorics of Groebner bases or via new methods proposed. Specific conjectures about the structure of the Cox rings of certain classes of varieties are proposed. The project also studies the Hilbert scheme of points in affine space, in particular the problem of describing the radical component (the closure of the set of radical ideals) and the singularities near monomial ideals.An algebraic variety is a space defined by the simultaneous vanishing of a collection of polynomial functions in several variables. A very fertile classical technique for studying the geometry of an algebraic variety is to consider the various natural coordinate systems that this variety admits. The Cox ring of an algebraic variety is a way to study all these coordinate systems simultaneously. The first objective of this project is to provide descriptions of the Cox rings of some algebraic varieties. The second objective of this proposal is to study the Hilbert scheme ofpoints: the space of all possible configurations of n points in d-dimensional space. The geometry of this set is rather intricate and the aim of the project is to describe all possible ways in which the n-points can collide into one another and the geometry of the Hilbert scheme near these collisions.