Rees algebras of ideals submaximally generated by quadrics

Abstract

The goal of this paper is to study the Rees algebra $\mathfrak{R}(I)$and the special fiber ring $\mathfrak{F}(I)$ for a family of ideals. Let $R=\mathbb{K}[x_1, \ldots, x_d]$ with $d\geq 4$ be a polynomial ring with homogeneous maximal ideal $\mathfrak{m}$. We study the $R$-ideals $I$, which are $\mathfrak{m}$-primary, Gorenstein, generated in degree 2, and have a Gorenstein linear resolution. In the smallest case, $d=4$, this family includes the ideals of $2\times 2$ minors of a general $3\times 3$ matrix of linear forms in $R$. We show that the defining ideal of the Rees algebra will be of fiber type. That is, the defining ideal of the Rees algebra is generated by the defining ideals of the special fiber ring and of the symmetric algebra. We use the fact that these ideals differ from $\mathfrak{m}^2$ by exactly one minimal generator to describe the defining ideal $\mathfrak{F}(I)$ as a sub-ideal of the defining ideal of $\mathfrak{F}(\mathfrak{m}^2)$, which is well known to be the ideal of $2\times 2$ minors of a symmetric matrix of variables.