Combinatorial problems in the theory of toric varieties

所提出的研究的一般领域是复曲面品种的组合学。复曲面簇是一些最简单的代数簇，一般理论可以在其上进行检验。由于复曲面簇是由组合数据定义的，因此这些簇的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。 复曲面品种的几何和组合之间的对应关系可以在两个方向上利用。几何问题可以简化为更简单的组合问题，相反，一些困难的组合问题有一个简单的几何证明。目前的建议着眼于两个这样的领域。斯坦利-莱斯纳环理论使用代数几何来研究复形中单形的数量。考克斯环理论用组合方法研究了Rees环的有限生成的交换代数问题。斯坦利-莱斯纳环理论的主要结果，证明了40年前，是庞加莱对偶和弱莱夫谢茨定理完全刻画了单纯多面体的所有可能的面数。我们建议研究这种对应关系的扩展及其亲属。该领域的主要问题是证明一般单纯球的弱Lefschetz定理，从而刻画这类球的所有面数。查尼-戴维斯猜想也涉及单纯球，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指数理论的目的是将单纯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标志编号。一个簇的最小模型程序可以完全用它的考克斯环来描述。如果这个簇的考克斯环是单生的，则称之为Mori Dream Space（MDS）.这些环首先研究了环面品种，他们有一个组合的描述。在这个建议中，我们感兴趣的第一个非平凡的情况下，一个加权的射影平面P（a，B，c）在一个点的爆破。这个简单的例子已经揭示了考克斯环的非常复杂的性质。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，其中考克斯环以符号幂的里斯环的名义出现。与组合和环面的方法，我们现在有能力大大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.