Image closure of symmetric wide-matrix varieties

TL;DR

This paper proves that the Zariski closure of the image of Sym(N)-equivariant maps between certain matrix and tensor schemes is finitely generated and stabilizes, revealing a symmetry-based Noetherian property.

Contribution

It establishes that the closures of these symmetric images are defined by finitely many orbits and are Sym(N)-Noetherian, advancing understanding of symmetry in algebraic geometry.

Findings

Zariski closure is finitely generated by symmetry orbits

The image closure is Sym(N)-Noetherian

Descending chains of symmetric closed subsets stabilize

Abstract

Let $X$ be an affine scheme of $k \times \mathbb{N}$-matrices and $Y$ be an affine scheme of $\mathbb{N} \times \cdots \times \mathbb{N}$-dimensional tensors. The group Sym$(\mathbb{N})$ acts naturally on both $X$ and $Y$ and on their coordinate rings. We show that the Zariski closure of the image of a Sym$(\mathbb{N})$-equivariant morphism of schemes from $X$ to $Y$ is defined by finitely many Sym$(\mathbb{N})$-orbits in the coordinate ring of $Y$. Moreover, we prove that the closure of the image of this map is Sym$(\mathbb{N})$-Noetherian, that is, every descending chain of Sym$(\mathbb{N})$-stable closed subsets stabilizes.