Combinatorial problems in the theory of toric varieties

提出的研究的一般领域是toric品种的组合。环面变异体是一些最简单的代数变异体，一般理论可以在其上进行检验。由于环面变体是由组合数据定义的，因此这些变体的所有几何问题都对应于涉及简单复合体、扇形、多面体的组合问题。环向变分的几何学和组合学之间的对应关系可以在两个方向上得到利用。几何问题可以简化为更简单的组合问题，相反，一些复杂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合体中简单体的数量。Cox环理论用组合方法研究了Rees环有限代的交换代数问题。40年前证明的Stanley-Reisner环理论的主要结果是，庞加莱对偶和弱Lefschetz定理完全描述了简单多面体的所有可能面数。我们建议研究这种对应关系及其相关关系的扩展。该领域的主要问题是证明一般简单球的弱Lefschetz定理，从而表征这种球的所有面数。Charney-Davis猜想也与简单球体有关，但增加了一个标志条件。该猜想在几何拓扑学中有重要的应用。cd指数理论旨在将简单球的所有结果推广到任意多面体球。这里的目标是描述所有这些球体的面和旗号。一个品种的最小模型程序可以完全用它的考克斯环来描述。如果其Cox环是有限生成的，则称为Mori梦空间（MDS）。这些环首先是研究环的品种，他们有一个组合的描述。在这个建议中，我们感兴趣的是第一种非平凡情况，即加权投影平面P（a,b,c）在一点上的膨胀。这个简单的例子已经揭示了考克斯环非常复杂的本质。该领域的主要问题是，这些爆炸中哪些是MDS。这个问题在交换代数中有很长的历史，Cox环以符号幂的Rees环的名义出现。利用组合和环面方法，我们现在有能力极大地简化和扩展以前的交换代数结果。

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.