Equations of the multi-Rees algebra of fattened coordinate subspaces

Abstract

In this paper, we describe the equations defining the multi-Rees algebra R[I1a1t1,...,Irartr] ( R = k[x1,...,xn]), where the ideals are generated by subsets of x1,...,xn. We also show that a family of binomials whose leading terms are squarefree, form a Gröbner basis for the defining equations with lexicographic order. We show that if we remove binomials that include x's, then remaining binomials form a Gröbner basis for the toric ideal associated to the multi-fiber ring. However binomials, including x's, in Gröbner basis of defining equations of the multi-Rees algebra are not necessarily defining equations of corresponding symmetric algebra. Despite this fact, we show that this family of ideals is of multi-fiber type.