On the h-vector of a matroid complex

This paper addresses the problem of characterizing the possible h-vectors of a matroid complex ∆. We define the h-vector in Section 1, both in terms of the f -vector, and as the dimension vector of a graded ring S(∆) which is given as the Stanley-Reisner ring of ∆ modulo a linear system of parameters. Section 1 ends with a brief survey of the literature on h-vectors of matroid complexes. In Section 2 we define a new ring T (∆), not always isomorphic to S(∆), and prove that it has the same dimension vector as S(∆). In Section 3 we consider the special case in which our matroid is given by an arrangement of hyperplanes in a rational vector space. In this case there is a natural choice of linear system of parameters such that the ring S(∆) is isomorphic to the cohomology ring of a certain variety, called a hypertoric variety, which can be constructed from the combinatorial data of the arrangement. Finally, in Section 4, we consider some explicit examples of the rings S(∆) and T (∆). The main result of this paper is based on a conjecture of Tamás Hausel. This paper was written for a class taught at Berkeley by Bernd Sturmfels, with guidance from Ezra Miller.