Course: Geometry of moduli of higher spin curves
The moduli space S_{g, r} of rspin curves parametrize rth order roots of the canonical bundles of curves of genus g. This space is an interesting cover of the moduli space of curves. For instance it carries a highly nontrivial virtual fundmental class whose numerical properties lead to a wellknown prediction of Witten. I will discuss various topics related to the birational geometry and intersection theory of these spaces, focusingboth on the more classical case of thetacharacteristics (r=2), as well as on the higher order analogues.
Course: Stable canonically polarized varieties
The theory of moduli of curves has been extremely successful and part of this success is due to the compactification of the moduli space of smooth projective curves by the moduli space of stable curves. A similar construction is desirable in higher dimensions but unfortunately the methods used for curves do not produce the same results in higher dimensions. In fact, even the definition of what "stable" should mean is not clear a priori. In order to construct modular compactifications of moduli spaces of higher dimensional canonically polarized varieties one must understand the possible degenerations that would produce this desired compactification that itself is a moduli space of an enlarged class of canonically polarized varieties. In this series of lectures I will start by discussing the difficulties that arise in higher dimensions and how theselead us to the definition of stable varieties and stable families. Time permitting construction of compact moduli spaces and recent relevant results will also be discussed.
