Universal moduli of parabolic sheaves on stable marked curves

Thesis

The topic of this thesis is the moduli theory of (parabolic) sheaves on stable curves. Using geometric invariant theory (GIT), universal moduli spaces of semistable parabolic sheaves on stable marked curves are constructed: `universal' indicates that these are moduli spaces of pairs where the underlying marked curve may vary as well as the parabolic sheaf (as in the Pandharipande moduli space for pairs of stable curves and torsion-free sheaves without augmentations).

As an intermediate step in this construction, we construct moduli spaces of semistable parabolic sheaves on flat families of arbitrary projective schemes (of any dimension or singularity type): this is the technical core of this thesis. These moduli spaces are projective, since they are constructed as GIT quotients of projective parameter spaces. The stability condition for parabolic sheaves depends on a choice of polarisation and is derived from the Hilbert-Mumford criterion. It is not quite the same as traditional stability with respect to parabolic Hilbert polynomials, but it is closely related to it, and the resulting moduli spaces are always compactifications of moduli of slope-stable parabolic sheaves. The construction works over algebraically closed fields of arbitrary characteristic.

Authors: Schlüeter, D; Dirk Schlüeter

• Publication date: 2011
• DOI: 10.5287/ora-dxejqvand
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: Oxford University, UK
• Language: English
• Keywords: geometric invariant theory; parabolic bundles; parabolic sheaves; algebraic geometry; moduli spaces; marked curves
• Subjects: Algebraic geometry; Mathematics; Geometry