Koszul multi-Rees algebras of principal L-Borel ideals

Abstract

Given a monomial m in a polynomial ring and a subset L of the variables of the polynomial ring, the principal L-Borel ideal generated by m is the ideal generated by all monomials which can be obtained from m by successively replacing variables of m by those which are in L and have smaller index. Given a collection I={I1,…,Ir} where Ii is Li-Borel for i=1,…,r (where the subsets L1,…,Lr may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets L1,…,Lr is chordal bipartite, then the defining equations of the multi-Rees algebra of I has a Gröbner basis of quadrics with squarefree lead terms under a lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gröbner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm.