Cohen-Macaulayness of Rees Algebras of Diagonal Ideals

TL;DR

This paper investigates the Cohen-Macaulay property of Rees algebras associated with diagonal ideals in determinantal rings, linking algebraic properties to geometric structures like join varieties.

Contribution

It establishes conditions under which the Rees algebra of diagonal ideals is Cohen-Macaulay, especially when it coincides with the symmetric algebra.

Findings

Rees algebra equals symmetric algebra under certain conditions

Rees algebra is Cohen-Macaulay when it coincides with the symmetric algebra

Connects algebraic properties to geometric structures like join varieties

Abstract

Given two determinantal rings over a field k. We consider the Rees algebra of the diagonal ideal, the kernel of the multiplication map. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety. When the Rees algebra and the Symmetric algebra coincide, we show that the Rees algebra is Cohen-Macaulay.