Rees Algebras of Diagonal Ideals

TL;DR

This paper proves that diagonal ideals of determinantal rings are of linear type and explores their Rees algebra structure, including special cases and their geometric interpretation as join varieties.

Contribution

It establishes that diagonal ideals in determinantal rings are of linear type and characterizes their Rees algebras and special fiber rings.

Findings

Diagonal ideals are of linear type in determinantal rings.

The defining ideal of the Rees algebra is characterized in certain cases.

The special fiber ring corresponds to the homogeneous coordinate ring of a join variety.

Abstract

There is a natural epimorphism from the symmetric algebra to the Rees algebra of an ideal. When this epimorphism is an isomorphism, we say that the ideal is of linear type. Given two determinantal rings over a field, we consider the diagonal ideal, the kernel of the multiplication map. We prove that the diagonal ideal is of linear type and recover the defining ideal of the Rees algebra in some special cases. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety.