Universal moduli of sheaves over curves and moduli of flags of varieties via Geometric Invariant Theory - Cooper, GE (2024)
This thesis is comprised of three main results. The first main result is the existence of GIT constructions for good moduli spaces of compactified universal Jacobians, and their higher rank analogues, over stacks of marked stable curves and stable maps. The construction is carried out in a way which allows for changes in stability conditions to be governed by variation of GIT. We also prove that the singularities of these good moduli spaces (when working over Mg,n) are canonical when g ≥ 4, extending work of Casalaina-Martin–Kass–Viviani. The second main result concerns the construction of quasi-projective coarse moduli spaces parametrising complete flags of subschemes of a fixed projective space P(V) up to projective automorphisms; these flags are obtained by intersecting non-degenerate non-singular subvarieties of P(V) of dimension n by flags of linear subspaces of P(V) of length n, with each positive dimension component of the flags being required to be non-singular and non-degenerate, and with the dimension 0 component being required to satisfy a Chow stability condition. These moduli spaces are constructed using non-reductive Geometric Invariant Theory, making use of a non-reductive analogue of quotienting-in-stages developed by Hoskins and Jackson. The final main result is the existence of quasi-projective fine moduli spaces of Gieseker unstable torsion-free coherent sheaves of uniform rank 1 on a reducible projective curve with Gorenstein singularities. These are the first examples of projective moduli spaces of unstable sheaves on projective schemes which admit fully modular descriptions, without any restriction on the Harder–Narasimhan length. These moduli spaces are constructed by taking iterated universal extension bundles over fine compactified Jacobians of the subcurves.