Tame graphs, clutters and their Rees algebras

TL;DR

This paper characterizes tame squarefree monomial ideals using combinatorial structures called clutters, linking algebraic properties to combinatorial configurations, and explores their implications for Rees algebras and fiber type properties.

Contribution

It provides a combinatorial characterization of tame squarefree monomial ideals and describes their algebraic and geometric properties, including their relation to Rees algebras.

Findings

A squarefree monomial ideal is tame iff its clutter is a union of isolated vertices and a complete d-partite d-uniform clutter.

Tame squarefree ideals are characterized by facets of their Stanley-Reisner complex having disjoint complements.

Tame squarefree ideals are of fiber type.

Abstract

A tame ideal is an ideal $I$ such that the blowup of the affine space $\mathbb{A}_k^n$ along $I$ is regular. In this paper, we give a combinatorial characterization of tame squarefree monomial ideals. More precisely, we show that a square free monomial ideal is tame if and only if the corresponding clutter is a union of some isolated vertices and a complete $d$-partite $d$-uniform clutter. It turns out that a squarefree monomial ideal is tame, if and only if the facets of its Stanley-Reisner complex have mutually disjoint complements. Also, we characterize all monomial ideals generated in degree at most 2 which are tame. Finally, we prove that tame squarefree ideals are of fiber type.