Topics

Perspectives on moduli of varieties (Kristin DeVleming)

In these lectures, I will introduce KSB(A) moduli spaces of varieties of (log) general type and K moduli spaces of Fano varieties.  We will survey KSB(A) stability and K stability in abstraction and with explicit examples.  Time permitting, we will also discuss wall-crossing for moduli spaces of pairs and several applications. 

The integral Hodge conjecture and higher K-theory (Elden Elmanto)

In this mini course, we survey the rich interaction between the birational geometry of an algebraic variety, cycles on them and their derived categories of coherent sheaves. This interaction is mediated by algebraic and topological K-theory - objects originating from algebraic topology. In particular I will demonstrate how one can prove the integral Hodge conjecture on some Fano 4 folds using these ideas. Anything original is with Nick Addington. (Elden has had to cancel for personal reasons)

Projectivity of Moduli (Jarod Alper)

With a view toward the moduli space of stable curves of genus g, we survey the techniques to prove projectivity. Beginning with a review of positivity properties of line bundles (ample, base point free, semiample, nef, and big), we will cover general projectivity criterion such as the Nakai--Moishezon Criterion and Kollár's Criterion for Ampleness.  We then present two constructions of Mgbar as a projective variety.  The first proof uses Kollár's Criterion and results in birational geometry while the second proof follows Mumford and Gieseker's GIT construction using Chow stability of a smooth curve.

Universality: the good, the bad, the ugly (Max Lieblich)

Moduli spaces solve universal problems. This is usually a source of joy, but it can also cause pain. In this lecture series, I will discuss the two main ways that things can go wrong. First, a moduli space can exhibit every finite-type singularity: this is usually called Murphy's Law (by Harris and Vakil). As I will explain, Vakil's key insight in studying this phenomenon was that geometric encodings of arithmetic through configurations of lines in the projective plane can be cleverly leveraged to create singularities in many moduli spaces. Second, a moduli stack can encode every way that an object can fail to descend to its field of moduli. In joint work with Dan Bragg, we made this second pathology precise with the idea of "gerbes of definition". As I will discuss, the universality of moduli theory creates a profusion of these gerbes of definition through a relative form of standard results in classical equivariant projective geometry.