Stillman's question for twisted commutative algebras

TL;DR

This paper constructs specific ideals in polynomial rings with group actions, demonstrating unbounded regularity and providing a negative answer to a generalization of Stillman's conjecture.

Contribution

It introduces a family of group-stable ideals in twisted commutative algebras and shows their regularity is unbounded, addressing a question related to Stillman's conjecture.

Findings

Constructed $ ext{GL}_m( ext{C})$-stable ideals in polynomial rings.

Proved the regularity of these ideals is unbounded.

Negatively answered a question on a generalization of Stillman's conjecture.

Abstract

Let $\mathbf{A}_{n, m}$ be the polynomial ring $\text{Sym}(\mathbf{C}^n \otimes \mathbf{C}^m)$ with the natural action of $\mathbf{GL}_m(\mathbf{C})$. We construct a family of $\mathbf{GL}_m(\mathbf{C})$-stable ideals $J_{n, m}$ in $\mathbf{A}_{n, m}$, each equivariantly generated by one homogeneous polynomial of degree $2$. Using the Ananyan-Hochster principle, we show that the regularity of this family is unbounded. This negatively answers a question raised by Erman-Sam-Snowden on a generalization of Stillman's conjecture.