Symbolic Powers of Cover Ideals and the Koszul Property
Joseph Skelton – Clemson University
Let G be a finite simple graph. This talk investigates the properties of symbolic powers of cover ideals associated to G. The results begin with combinatorially changing a given graph by whiskering (introducing a degree one vertex) a cycle cover of the graph. We show that this process forces all symbolic powers of the cover ideal J(G) to be Koszul or, equivalently, componentwise linear. We follow this result with a method of gluing two graphs, with Koszul cover ideals, together to obtain a new graph whose cover ideal, and all its symbolic powers, are Koszul.
We then move to studying a spanning bipartite subgraph, B_G, of G. We first give conditions on J(G) which give that symbolic power J(G)^{(s)} is not componentwise linear for s>2. We then use this result and B_G to give necessary and sufficient conditions on J(G) such that J(G)^{(s)} is componentwise linear for all s>1. In particular when G is a graph such that G\ N[A] has a simplicial vertex for any independent set A, or when G is a bipartite graph.
This is joint work with Yan Gu, Tài Hà, and S. Selvaraja.