Combinatorial problems in the theory of toric varieties

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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.