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

本计画研究交换代数、组合数学与代数几何之间的两个共同点：代数簇的考克斯环与点的希尔伯特图式。代数簇X$的考克斯环是一个多重分次环，其GIT等价物是X$通过有理映射的象的坐标环.对于许多有趣的变种，所谓的Mori Dream空间，考克斯环是一个生成的k-代数。该项目旨在开发工具，以有效地实现某些Mori Dream空间的考克斯环，在显式坐标和定义方程方面。这样的描述是通过Koszul过滤，通过组合Groebner基地或通过新的方法提出。给出了某些簇的考克斯环的结构的具体刻划. 该项目还研究了仿射空间中点的希尔伯特方案，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的多项式函数集合同时消失定义的空间。研究代数簇的几何的一个非常丰富的经典技术是考虑这个簇所允许的各种自然坐标系。代数簇的考克斯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提供一些代数簇的考克斯环的描述。这个建议的第二个目标是研究希尔伯特方案的点：空间的所有可能的配置n点在d维空间。这个集合的几何形状相当复杂，该项目的目的是描述所有可能的方式，其中n点可以相互碰撞，以及这些碰撞附近的希尔伯特方案的几何形状。

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.