Good Hilbert functors

TL;DR

This paper introduces the good Hilbert functor, proving its algebraicity and generalizing several Hilbert moduli problems, while also extending a formal GAGA result for good moduli spaces.

Contribution

It defines the good Hilbert functor, proves its algebraic nature, and generalizes existing results on Hilbert schemes and formal GAGA for good moduli spaces.

Findings

The good Hilbert functor is algebraic.

Generalization of multigraded and invariant Hilbert schemes.

Extension of formal GAGA results for good moduli spaces.

Abstract

We introduce the good Hilbert functor and prove that it is algebraic. This functor generalizes various versions of the Hilbert moduli problem, such as the multigraded Hilbert scheme and the invariant Hilbert scheme. Moreover, we generalize a result concerning formal GAGA for good moduli spaces.