A blow-up construction and graph coloring

Abstract

Given a graph G (or more generally a matroid embedded in a projective space), we construct a sequence of algebraic varieties whose geometry encodes combinatorial information about G. For example, the chromatic polynomial of G can be computed as an intersection product of certain classes on these varieties, or recovered in terms of the Segre classes of related subschemes of P n; other information such as Crapo's invariant also finds a very natural geometric counterpart. The note presents this construction, and gives ‘geometric’ proofs of a number of standard combinatorial results on the chromatic polynomial and Crapo's invariant.