Quotients for non-reductive group actions and applications to moduli spaces of matrix factorisations

Abstract

Classical geometric invariant theory as developed by D. Mumford in his book "Geometric Invariant Theory" only applies to actions of a reductive group. G. Bérczi, B. Doran, T. Hawes, and F. Kirwan construct in "Projective Completions of graded unipotent quotients" quotients for non-reductive group actions on projective and irreducible schemes under the assumption that the unipotent radical admits a positive grading.

In the first part of this thesis, we study actions of a non-reductive group G on a separated scheme X of finite type over the base field. We formulate a definition for semi-stable and stable points with respect to a pair (K,L) of two G-linearisations and a chosen Levi-factor of G. If the line bundle K is ample, then the locus of semi-stable points admits a good quotient which contains a geometric quotient of the locus of stable points as an open subset. We give a sufficient condition such that the good quotient of the locus of semi-stable points is projective. Further, we prove a Hilbert-Mumford-style criterion to compute the set of semi-stable points. This generalises results by G. Bérczi, B. Doran, T. Hawes and F. Kirwan.

In the second part of this thesis, we apply the results of non-reductive geometric invariant theory to construct compactifications of moduli spaces of matrix factorisations of Shamash type. We examine two cases in particular. Let an elliptic quintic curve or a twisted quartic curve be contained in a cubic threefold which is cut out by a homogeneous form f of degree three. We obtain a matrix factorisation of f from the minimal resolution of the homogeneous coordinate ring of the curve over the homogeneous coordinate ring of the ambient projective space with a construction by J. Shamash. For both types of matrix factorisations, we give sufficient numerical conditions on the chosen linearisations (K,L) such that the good quotient of semi-stable generalised matrix factorisations by the action of the automorphism group is projective. This quotient contains a geometric quotient of the locus of stable matrix factorisations as an open subset.