Abstract:
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 methodsused 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 these lead 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.
Geometry of moduli of
higher spin curves (Farkas)
Abstract:
The
moduli
space S_{g, r} of r-spin curves parametrize r-th 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 non-trivial virtual fundmental class whose numerical
properties lead
to a well-known prediction of Witten. I will discuss various topics
related to the birational geometry and intersection theory of these
spaces, focusing both on the more classical case of
theta-characteristics (r=2), as well as on the higher order analogues.
Each of the courses will consist of 4 lectures to be
held Monday 5 to Thursday 8. The first session will be held
Monday 5 at 14h00m.
Timetable:
to be defined
Location:
Departamento
de
Matemática,
Instituto
Superior
Técnico
Post
Graduation
Building