Knutson ideals of generic matrices

TL;DR

This paper proves that determinantal ideals of generic matrices are Knutson ideals, enabling a new approach to Gr"obner bases for sums of such ideals, which simplifies computations in algebraic geometry.

Contribution

It establishes that determinantal ideals are Knutson ideals and shows how to construct Gr"obner bases for sums of these ideals from individual bases.

Findings

Determinantal ideals of generic matrices are Knutson ideals.

Union of Gr"obner bases of component ideals forms a Gr"obner basis of their sum.

Simplifies Gr"obner basis computations for sums of determinantal ideals.

Abstract

In this paper we show that determinantal ideals of generic matrices are Knutson ideals. This fact leads to a useful result about Gr\"obner bases of certain sums of determinantal ideals. More specifically, given $I=I_1+\ldots+I_k$ a sum of ideals of minors on adjacent columns or rows, we prove that the union of the Gr\"obner bases of the $I_j$'s is a Gr\"obner basis of $I$.