On a conjecture of Vasconcelos via Sylvester forms

We study 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 and that the additional monomial has the property that all variables 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 providing an affirmative partial answer to a conjecture of W. Vasconcelos.