Rees algebra and almost linearly presented ideals in three variables

TL;DR

This paper derives explicit formulas for the defining ideal of the Rees algebra of certain height two perfect ideals in three variables, characterized by almost linear presentations and specific rank conditions.

Contribution

It provides new explicit formulas for the Rees algebra of almost linearly presented ideals in three variables under particular rank and G-s conditions.

Findings

Explicit formulas for the Rees algebra's defining ideal.

Characterization of ideals satisfying specific G-s conditions.

Analysis of ideals with almost linear presentations in three variables.

Abstract

Let $R=\k[x,y,z]$ and $I=(f_0,\dots,f_{n-1})$ be a height two perfect ideal which is almost linearly presented (that is, all but the last column have linear entries, but the last column has entries which are homogeneous of degree $2$). Further we suppose that after modulo an ideal generated by two variables, the presentation matrix has rank one. Also, the ideal $I$ satisfies $\Gs{{2}}$ but not $\Gs{3}$, then we obtain explicit formulas for the defining ideal of the Rees algebra $\rees(I)$ of $I$.