Rees Algebras and Almost Linearly Presented Ideals

TL;DR

This paper investigates the structure of Rees algebras for a class of grade 2 perfect ideals with almost linear presentation matrices, providing explicit descriptions and introducing iterated Jacobian duals.

Contribution

It offers explicit descriptions of the defining ideals of Rees algebras for almost linearly presented ideals and introduces the concept of iterated Jacobian duals.

Findings

Explicit forms of the defining ideal of the Rees algebra for the class of ideals studied.

Introduction of the notion of iterated Jacobian duals.

Enhanced understanding of the algebraic structure of Rees algebras in this context.

Abstract

Consider a grade 2 perfect ideal $I$ in $R=k[x_1,\cdots,x_d]$ which is generated by forms of the same degree. Assume that the presentation matrix $\varphi$ is almost linear, that is, all but the last column of $\varphi$ consist of entries which are linear. For such ideals, we find explicit forms of the defining ideal of the Rees algebra $\mathcal{R}(I)$. We also introduce the notion of iterated Jacobian duals.