Rees algebras of square-free monomial ideals

TL;DR

This paper investigates the defining equations and relation type of Rees algebras of square-free monomial ideals, providing bounds, explicit descriptions, and conditions for linearity based on the ideal's generator graph structure.

Contribution

It establishes bounds on the relation type for certain square-free monomial ideals and introduces the generator graph as a tool to determine when such ideals are of linear type.

Findings

Relation type at most n-2 for n ≤ 5 when generators are of same degree

Explicit defining equations for the Rees algebra in general cases

Identification of classes of ideals of linear type based on generator graph structure

Abstract

We study the defining equations of the Rees algebra of square-free monomial ideals in a polynomial ring over a field. We determine that when an ideal $I$ is generated by $n$ square-free monomials of the same degree then $I$ has relation type at most $n-2$ as long as $n \leq 5$. In general, we establish the defining equations of the Rees algebra in this case. Furthermore, we give a class of examples with relation type at least $n-2$, where $n=\mu(I)$. We also provide new classes of ideals of linear type. We propose the construction of a graph, namely the generator graph of an ideal, where the monomial generators serve as vertices for the graph. We show that when $I$ is a square-free monomial ideal generated in the same degree that is at least 2 and the generator graph of $I$ is the graph of a disjoint union of trees and graphs with unique odd cycles then $I$ is an ideal of linear type.