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arXiv · 2007.03038

Stillman's question for twisted commutative algebras

Karthik Ganapathy
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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.

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Karthik Ganapathy. 2020-07-06. Stillman's question for twisted commutative algebras. https://doi.org/10.1216/jca.2022.14.315

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