Stillman’s question for twisted commutative algebras

Abstract

Let 𝔸n,m be the polynomial ring Sym(ℂn⊗ℂm) with the natural action of GLm(ℂ). We consider a family of GLm(ℂ)-stable ideals Jn,m in 𝔸n,m, each equivariantly generated by one homogeneous polynomial of degree 2 and show that the regularity of this family is unbounded. Using this, we negatively answer a question raised by Erman, Sam and Snowden on a generalization of Stillman’s conjecture.