Authors:	Buhr, Lukas
Advisor:	Lehn, Manfred van Straten, Duco
Title:	Quotients for non-reductive group actions and applications to moduli spaces of matrix factorisations
Online publication date:	15-Feb-2024
Year of first publication:	2024
Language:	english
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.
DDC:	510 Mathematik 510 Mathematics
Institution:	Johannes Gutenberg-Universität Mainz
Department:	FB 08 Physik, Mathematik u. Informatik
Place:	Mainz
ROR:	https://ror.org/023b0x485
DOI:	http://doi.org/10.25358/openscience-9985
URN:	urn:nbn:de:hebis:77-openscience-24eb48c4-9385-4805-858a-1f96686fa4c77
Version:	Original work
Publication type:	Dissertation
License:	In Copyright
Information on rights of use:	http://rightsstatements.org/vocab/InC/1.0/
Extent:	xiii, 81 Seiten
Appears in collections:	JGU-Publikationen