Combinatorial problems in the theory of toric varieties

环曲面簇理论中的组合问题

• 批准号: RGPIN-2020-04335
• 负责人: Karu, Kalle
• 金额: $ 1.75万
• 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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716980
• 关键词: Combinatorial; problems; theory; toric; varieties

The general area of the proposed research is the combinatorics of toric varieties. Toric varieties are some of the simplest algebraic varieties on which general theories can be tested. Since toric varieties are defined by combinatorial data, all problems in the geometry of these varieties correspond to combinatorial problems that involve simplicial complexes, fans, polytopes. The correspondence between geometry and combinatorics of toric varieties can be exploited in both directions. Geometric problems can be reduced to simpler combinatorial ones, and conversely, some hard combinatorics problems have a simple geometric proof. The current proposal looks at two such areas. The Stanley-Reisner ring theory uses algebraic geometry to study the number of simplices in a complex. The theory of Cox rings uses combinatorial methods to study the commutative algebra problem of finite generation of a Rees ring. The main result in the theory of Stanley-Reisner rings, proved 40 years ago, is that Poincare duality and the Weak Lefschetz theorem completely characterize all possible face numbers of simplicial polytopes. We propose to study the extension of this correspondence and its relatives. The main problem in the area is to prove the Weak Lefschetz theorem for general simplicial spheres and thus characterize all face numbers of such spheres. The Charney-Davis conjecture also concerns simplicial spheres, but with an added flag condition. The conjecture has important applications in geometric topology. The theory of the cd-index aims at generalizing all results from simplicial spheres to arbitrary polyhedral spheres. The goal here is to characterize the face and flag numbers of all such spheres. The minimal model program for a variety can be completely described in terms of its Cox ring. The variety is called a Mori Dream Space (MDS) if its Cox ring is finitely generated. These rings were first studied for toric varieties where they have a combinatorial description. In this proposal we are interested in the first nontrivial case, the blowup of a weighted projective plane P(a,b,c) at a point. This simple case already brings out the very complicated nature of Cox rings. The main question in the area is which of these blowups are MDS. The problem has a long history in commutative algebra where Cox rings appear under the name of Rees rings of symbolic powers. With combinatorial and toric methods we now have the ability to greatly simplify and extend previous commutative algebra results.

所提出的研究的一般领域是环簇的组合学。Toric簇是可以用来检验一般理论的最简单的代数簇。由于环簇是由组合数据定义的，所以环簇几何中的所有问题都对应于涉及单纯复形、扇形、多面体的组合问题。 环面簇的几何学和组合学之间的对应关系可以在两个方向上进行探索。几何问题可以归结为更简单的组合问题，相反，一些难的组合问题有一个简单的几何证明。目前的提案着眼于两个这样的领域。Stanley-Reisner环理论使用代数几何来研究复数中的单数。Cox环理论利用组合方法研究Rees环的有限生成的交换代数问题。 40年前证明的Stanley-Reisner环理论的主要结果是Poincare对偶和弱Lefschetz定理完全刻画了单纯多面体的所有可能面数。我们建议研究这一对应及其相关函数的推广。该领域的主要问题是证明一般单纯球面的弱Lefschetz定理，从而刻画这类球面的所有面数。查尼-戴维斯猜想也涉及单纯球，但有一个附加的旗子条件。该猜想在几何拓扑学中有重要的应用。CD指数理论旨在将单纯球面上的所有结果推广到任意多面体球面上。这里的目标是描述所有这类球体的面和旗帜编号。 一个变种的最小模型规划可以完全用它的COX环来描述。这种簇称为Mori Dream Space(MDS)，如果它的Cox环是有限生成的。这些环首先被研究为环簇，其中它们具有组合描述。在这个建议中，我们感兴趣的是第一种非平凡的情形，即加权射影平面P(a，b，c)在一点的爆破。这个简单的例子已经揭示了Cox环的非常复杂的性质。该地区的主要问题是，这些爆炸事件中哪些是MDS。这个问题在交换代数中有很长的历史，其中Cox环以符号幂的Rees环的名义出现。有了组合和环面方法，我们现在有能力极大地简化和推广以前的交换代数结果。