Renzo Cavalieri (Colorado State University)
Title: An exploration of the moduli spaces of curves.
Abstract: This mini-course is meant to be an introduction to various
fundamental ideas in moduli spaces theory, through the concrete
example of the moduli space of curves. The essential idea is that the
geometric structure (i.e. the extra structure one gives to a set of
points to turn it into a space, be it a complex manifold, an algebraic
variety, scheme etc) of a moduli space is phrased in terms of a
functor that describes the moduli problem. This establishes a working
dictionary between the geometry of the moduli space and the geometry
of families of objects that one wishes to parameterize. We will use
this dictionary to understand the basic geometry of the moduli spaces
of curves (for example the boundary stratification for the natural
compactification to stable curves), to construct natural bundles on
the moduli spaces of curves, and to explore the intersection theory
of their characteristic classes.
Izzet Coskun (University of Illinois at Chicago)
Title: The birational geometry of the moduli spaces of curves.
Abstract: In these lectures, we will discuss the birational geometry
and Mori theory of moduli spaces such as the moduli space of stable
curves of genus g and the Kontsevich moduli space of genus zero stable
maps. We will begin by studying the ample and effective cones of these
moduli spaces. We will study the stable base locus decomposition of
the effective cone and the corresponding birational models. We will
discuss recent progress in the minimal model program for the moduli
space of curves due to Hassett and Hyeon. The focus of these lectures
will be to illustrate the general theory in a few concrete, explicit
A list of references can be found
Emanuele Macri (University of Utah)
Title: An introduction to Hilbert Schemes.
In this series of four lectures we will study the construction and
some basic properties of the Hilbert schemes.
In the first lecture we will present the basic idea through several examples.
In the second and third lecture we will concentrate on the more formal
aspects of the construction.
The deformation theory for the Hilbert scheme will be the subject of
the last lecture.
Further topics will be discussed in the exercise sessions.
James McKernan (Massachusetts Institute of Technology)
Title: An introduction to the minimal model program.
In these lectures we will cover some of the basics of the minimal model program, such as Kawamata-Viehweg vanishing, bend and break, the cone theorem, finite generation of the canonical ring, the MMP with scaling and finiteness