Compactifications of Universal Moduli Spaces of Vector Bundles and the Log-Minimal Model Program on $\overline{M}_{g}$

Abstract

Recent work on the log-minimal model program for the moduli space of curves, as well as past results of Caporaso, Pandharipande, and Simpson motivate an investigation of compactifications of the universal moduli space of slope semi-stable vector bundles over moduli spaces of curves arising in the Hassett-Keel program. Our main result is the construction of a compactification of the universal moduli space of vector bundles over several of these moduli spaces, along with a complete description in the case of pseudo-stable curves.Theorem 1.0.1. For all α ∈ Q ∩ (2/3, 1], e, r ≥ 1 and g ≥ 3, there exists a projective variety U e,r,g (α) with a canonical projection π : U e,r,g (α) → M g (α) such that the diagram U e,r (C • g /M • g ) / / U e,r,g (α)

The Picard group of the universal moduli space of vector bundles on stable curves

“…over the Hyeon-Morrison's moduli stack M wp g of weakly-pseudo-stable curves). In higher rank, the conjectural first step of the LMMP for the Pandharipande's compactification U r,d,g has been described by Grimes in [22]: using the torsion free approach, he constructs a compactification U ps r,d,g of the moduli space of slope-semistable vector bundles over M ps g . In order to construct birational compact models for the Pandharipande compactification of U r,d,g , it is useful to have an explicit description of its rational Picard group which naturally embeds into the rational Picard group of the moduli stack T F ss r,d,g of slope-semistable torsion free sheaves over stable curves.…”