Equations of the multi-Rees algebra of fattened coordinate subspaces

TL;DR

This paper characterizes the defining equations of the multi-Rees algebra for fattened coordinate subspaces, providing Gröbner bases and exploring their properties related to toric ideals and fiber rings.

Contribution

It introduces a family of binomials forming a Gröbner basis for the multi-Rees algebra's defining equations and analyzes their relation to symmetric and toric ideals.

Findings

A Gröbner basis with lex order is identified for the defining equations.

Removing certain binomials yields a Gröbner basis for the associated toric ideal.

The family of ideals is shown to be of multi-fiber type.

Abstract

In this paper we describe the equations defining the multi-Rees algebra $k[x_1,\dots,x_n][I_1^{a_1}t_1,\dots,I_r^{a_r}t_r]$, where the ideals are generated by subsets of $x_1,\dots,x_n$. We also show that a family of binomials whose leading terms are squrefree, form a Gr\"{o}bner basis for the defining equations with lexicographic order. We show that if we remove binomials that include $x$'s, then remaining binomials form a Gr\"{o}bner basis for the toric ideal associated to the multi-fiber ring. However binomials, including $x$'s, in Gr\"{o}bner 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.