Stack structures on GIT quotients parametrizing hypersurfaces

TL;DR

This paper introduces a stack structure on GIT quotients parametrizing hypersurfaces, revealing intrinsic geometric information in invariant rings that is not captured by traditional Proj constructions.

Contribution

It proposes a novel stack-theoretic approach to GIT quotients, enhancing the understanding of their geometric properties, especially for binary forms of low degree.

Findings

Stack structure encodes additional geometric information.

Analysis of classical invariant rings through the stack perspective.

Potential applications to understanding hypersurface moduli.

Abstract

We suggest to endow Mumford's GIT quotient scheme with a stack structure, by replacing Proj(-) of the invariant ring with its stack theoretic analogue. We analyse the stacks resulting in this way from classically studied invariant rings, and in particular for binary forms of low degree. Our viewpoint is that the stack structure carries interesting geometric information that is intrinsically present in the invariant ring, but lost when passing to its Proj(-).