Radical generic initial ideals

TL;DR

This paper surveys Cartwright-Sturmfels ideals, highlighting their radical multigraded generic initial ideals and unique Borel-fixed point in their Hilbert scheme, with new classifications among determinantal and Schubert ideals.

Contribution

It provides a comprehensive survey of Cartwright-Sturmfels ideals and introduces a new class, characterizing certain determinantal and Schubert ideals as Cartwright-Sturmfels.

Findings

Cartwright-Sturmfels ideals have a unique Borel-fixed point in their Hilbert scheme.

Certain determinantal and Schubert ideals are characterized as Cartwright-Sturmfels.

The paper discusses properties of universal Gröbner bases and initial ideals for these ideals.

Abstract

In this paper, we survey the theory of Cartwright-Sturmfels ideals. These are Z^n-graded ideals, whose multigraded generic initial ideal is radical. Cartwright-Sturmfels ideals have surprising properties, mostly stemming from the fact that their Hilbert scheme only contains one Borel-fixed point. This has consequences, e.g., on their universal Groebner bases and on the family of their initial ideals. In this paper, we discuss several known classes of Cartwright-Sturmfels ideals and we find a new one. Among determinantal ideals of same-size minors of a matrix of variables and Schubert determinantal ideals, we are able to characterize those that are Cartwright-Sturmfels.