On a conjecture of Vasconcelos

TL;DR

This paper proves that the Rees algebra of certain monomial ideals is almost Cohen--Macaulay, providing explicit generators and a quasi-homogeneous structure, thus confirming a case of Vasconcelos's conjecture.

Contribution

It establishes the almost Cohen--Macaulay property of the Rees algebra for a class of monomial ideals and describes its explicit Sylvester form generators.

Findings

Rees algebra has a quasi-homogeneous structure

Presentation ideal generated by Sylvester forms

Rees algebra is almost Cohen--Macaulay

Abstract

One studies the structure of the Rees algebra of an almost complete intersection monomial ideal of finite co-length in a polynomial ring over a field, assuming that the least pure powers of the variables contained in the ideal have the same degree. It is shown that the Rees algebra has a natural quasi-homogeneous structure and its presentation ideal is generated by explicit Sylvester forms. A consequence of these results is a proof that the Rees algebra is almost Cohen--Macaulay, thus answering affirmatively an important case of a conjecture of W. Vasconcelos.