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