UC Berkeley

Matroids are combinatorial abstractions of hyperplane arrangements, and have been a bridge for fruitful interactions between combinatorics and algebraic geometry. In particular, the recent development of the Hodge theory of matroids in [AHK18] showed that the Chow ring of a matroid satisfies properties enjoyed by cohomology rings of smooth complex projective varieties. Namely, these are the Poincare duality property, the hard Lefschetz property, and the Hodge-Riemann relations. The validity of these properties resolved several major conjectures in matroid theory.

In this thesis, we introduce a presentation of the Chow ring of a matroid by a new set of generators, called "simplicial generators." These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation via the theory of matroid quotients. Using this combinatorial interpretation, we (i) produce a bijection between a monomial basis of the Chow ring and a relative generalization of Schubert matroids, (ii) recover the Poincare duality property, (iii) give a formula for the volume polynomial, which we show is log-concave in the positive orthant, and (iv) recover the validity of Hodge-Riemann relations in degree 1, which is the part of the Hodge theory of matroids that currently accounts for all combinatorial applications of [AHK18]. Our work avoids the use of "flips," which is the key technical tool employed in [AHK18]. We also apply the tools developed here to study two particular divisor classes motivated by the geometry of wonderful compactifications of hyperplane arrangement complements.