School on birational geometry and moduli spaces
June 1-11, 2010 at the University of Utah






MRC 2010 - Snowbird



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 examples. A list of references can be found here
Emanuele Macri (University of Utah)
Title: An introduction to Hilbert Schemes.

Abstract: 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.

Abstract: 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 of models.