Additions to a theorem of Morey-Ulrich

TL;DR

This paper extends a theorem by Morey and Ulrich concerning perfect ideals of codimension 2 in polynomial rings, specifically addressing cases where certain generic conditions are partially satisfied.

Contribution

It provides an extended formulation of the Morey-Ulrich theorem for perfect ideals satisfying condition G_{d-1} but not G_d.

Findings

Extended theorem applicable under weaker conditions

Characterization of perfect ideals with specific presentation matrices

New insights into the structure of such ideals

Abstract

Let $R = k[x_1,\ldots, x_d]$ denote a standard graded polynomial ring over an algebraically closed field $k$, and let $I \subset R$ be a perfect ideal of codimension $2$ with an $n\times (n-1)$ linear presentation matrix $\phi$. We prove an extended formulation of a theorem of Morey and Ulrich in the case where $I$ satisfies condition $G_{d-1}$, but not condition $G_d$.