The geometry of polynomial representations

TL;DR

This paper introduces the concept of GL-varieties, infinite-dimensional algebraic varieties with group actions, and explores their foundational properties, aiding in understanding invariants and proving conjectures in algebraic geometry.

Contribution

It systematically studies GL-varieties, establishing foundational results like a Chevalley-type theorem, advancing the theoretical framework for infinite-dimensional algebraic varieties.

Findings

Defined GL-varieties and their properties

Proved a Chevalley-type theorem for GL-varieties

Facilitated studies of invariants like tensor rank and strength

Abstract

We define a GL-variety to be a (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial representation. Such varieties have been used to study asymptotic properties of invariants like strength and tensor rank, and played a key role in two recent proofs of Stillman's conjecture. We initiate a systematic study of GL-varieties, and establish a number of foundational results about them. For example, we prove a version of Chevalley's theorem on constructible sets in this setting.