Resumen de Regularity and gröbner bases of the rees algebra of edge ideals of bipartite graphs

Yairon Cid Ruiz

• Let G be a bipartite graph and I = I(G) be its edge ideal. The aim of this note is to investigate different aspects of the Rees algebra R(I) of I. We compute its regularity and the universal Gröbner basis of its defining equations; interestingly, both of them are described in terms of the combinatorics of G. We apply these ideas to study the regularity of the powers of I. For any s ≥ match(G)+ |E(G)|+1 we prove that reg(I s+1) = reg(I s )+2 and that for an s ≥ 1 we have the inequality reg(I s ) ≤ 2s + match(G) − 1.