Stillman's question for twisted commutative algebras

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.

4 pages; added funding information and minor edits

References in corpus (4)

• Introduction to twisted commutative algebras
• Big polynomial rings and Stillman's conjecture
• GL-equivariant modules over polynomial rings in infinitely many variables. II
• Generalizations of Stillman's conjecture via twisted commutative algebras