Algebraic Geometry Seminar

For $C$ a smooth projective curve, and $G$ a simple, simply connected complex group, let $M_C(G)$ be the moduli space of semistable $G-$principal bundles on $C$. As the curve $C$ moves in the moduli $\mathcal{M}_g$ of smooth curves, the spaces $M_C(G)$ are known to define a flat family of schemes, and this family can be extended to the Deligne-Mumford compactification $\bar{\mathcal{M}}_g$. We describe the geometry of the fibers of this family which appear at the stable boundary, in particular we discuss a recent result which shows that the fibers over maximally singular curves contain an important and ubiquitous moduli space, the free group character variety $\mathcal{X}(F_g, G),$ as a dense, open subspace. This latter space is a moduli space of representations of $F_g$ in $G$, and naturally appears as an object of interest in Teichm\"uller theory, the theory of geometric structures, and the theory of Higgs bundles. We will also discuss the structure of the coordinate ring of $M_C(G)$, and features of the resulting compactification of the character variety when $G = SL_2(\C)$.