Algebraic geometry of toric varieties and its applications to combinatorics.

环面簇的代数几何及其在组合数学中的应用。

• 批准号: RGPIN-2015-05787
• 负责人: Karu, Kalle
• 金额: $ 1.02万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675380
• 关键词: Algebraic; geometry; toric; varieties; its

We study problems originating in the geometry of toric varieties. Toric varieties are geometric objects that are defined combinatorially by a convex polytope or a fan. The geometric properties of a toric variety are therefore reflected in the combinatorics of the polytope or fan. ***The theory of toric varieties has applications in both algebraic geometry and combinatorics. In algebraic geometry toric varieties are considered as being much simpler than general varieties, so hard conjectures can first be tested and proved on toric varieties. Examples of such results first proved for toric varieties include the minimal model program, mirror symmetry, and others.****Toric varieties also have applications in combinatorics. These typically involve interpreting some combinatorial property of a polytope or a fan geometrically. The most well-known example of this is the description of how many faces of each dimension a simple polytope can have. These face numbers are described in terms of cohomology of the toric variety, and the relations coming from geometry give all the necessary conditions on these numbers. ***The first part of this proposal relates to the combinatorics of lattice points in a polytope. The geometric counterpart of this is the study of Cox rings of toric varieties and their blowups. The remaining parts of the proposal study the number of faces or chains of faces of a polytope. This in algebraic geometry corresponds to studying various cohomology theories, such as intersection cohomology, cd-index, algebraic cobordism theory and others.**

我们研究起源于环簇几何的问题。圆环簇是由凸多面体或扇形组合定义的几何对象。因此，环面簇的几何性质反映在多面体或扇形的组合中。*环簇理论在代数几何和组合学中都有应用。在代数几何中，环簇被认为比一般簇简单得多，因此困难的猜想可以首先在环簇上得到检验和证明。最早被证明为Toric簇的例子包括极小模型程序、镜像对称性等。Toric簇在组合学中也有应用。这些通常涉及从几何上解释多面体或扇形的某些组合性质。最著名的例子是描述一个简单的多面体可以有多少个每个维度的面。这些面数用环面簇的上同调来描述，来自几何的关系给出了关于这些数的所有必要条件。*本建议的第一部分涉及多面体中格点的组合。与此对应的是环簇的Cox环及其爆破的研究。提案的其余部分研究多面体的面数或面链。在代数几何中，这相当于学习各种上同调理论，如交上同调、CD-指标、代数余边理论等。