Title: Betti numbers of secant powers of the edge ideal of a graph
Speaker: Reza Akhtar, Miami University (Ohio)
Seminar Type: Algebraic Geometry
Abstract: The edge ideal of a graph, first defined by R. Villareal, is an ideal in a certain polynomial ring over a field, which encodes the combinatorial structure of the graph. Being a finitely generated module over a graded ring, it is natural to study the (graded) Betti numbers of this ideal, and this topic has received considerable attention in the literature. Recently, there has been some interest on the part of algebraic geometers in computing Betti numbers of the higher secant powers of the edge ideal; these are the ideals which define the secant varieties corresponding to the variety defined by the edge ideal. This talk will focus on a recursive formula which gives the Betti numbers of (the secant powers of the edge ideal of) the join of a graph with an independent set, in terms of the Betti numbers of the original graph. This formula will be applied to compute all the Betti numbers in the case of complete graphs and complete multipartite graphs, recovering earlier results of Jacques.