Non-Reductive GIT and Moduli of Unstable Objects

George Cooper (University of Oxford)

Tuesday 14th November, 2023 16:00-17:00 Maths 110

Abstract

Geometric Invariant Theory (GIT) is a powerful theory for producing quotients within algebraic geometry, with important applications to moduli theory.

Given a reductive group G acting on a projective variety X, GIT produces an open semistable locus and a projective quotient of this locus. The complement of the semistable locus, the unstable locus, admits a stratification by “instability type”. In order to form quotients of these unstable strata, one needs to consider how to carry out GIT for actions of non-reductive groups.

I will explain how recent results in GIT, due to Bérczi, Doran, Hawes and Kirwan, allow for the formation of quotients by actions of non-reductive groups in certain cases. I will then explain how non-reductive GIT can be used to construct (universal) quasi-projective moduli spaces of unstable rank 1 sheaves on singular curves, which in many cases are projective.

No prior knowledge of GIT will be assumed.

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