On Rees algebras of 2-determinantal ideals

TL;DR

This paper proves that the Rees algebra and special fiber ring of 2-determinantal ideals are Cohen-Macaulay and Koszul, using a novel stratification approach and Groebner degenerations.

Contribution

It introduces a new method analyzing a stratification of the Hilbert scheme to study Rees algebras of determinantal ideals, establishing their Cohen-Macaulay and Koszul properties.

Findings

Rees algebra of 2-determinantal ideals is Cohen-Macaulay.

Rees algebra of 2-determinantal ideals is Koszul.

Rees algebra and special fiber ring are quadratic algebras.

Abstract

Let I be the ideal of minors of a 2 by n matrix of linear forms with the expected codimension. In this paper we prove that the Rees algebra of I and its special fiber ring are Cohen-Macaulay and Koszul; in particular, they are quadratic algebras. The main novelty in our approach is the analysis of a stratification of the Hilbert scheme of determinantal ideals. We study degenerations of Rees algebras along this stratification, and combine it with certain squarefree Groebner degenerations.