The Birational Geometry of Moduli Spaces

Abstract

The purpose of these lecture notes is to introduce the basics of the birational geometry of moduli spaces to students who have taken an introductory course in algebraic geometry. We concentrate on a few key ideas and examples. We define the cones of ample and effective divisors, compute them for a few examples such as the blowup of $\mathbb{P}^2$ at one or two points. Then we discuss the ample and effective cones of the Hilbert scheme of points on $\mathbb{P}^2$. Finally, in the last section, we give a guide to the literature on other moduli spaces. These are the notes for two lectures that I delivered at the CIMPA/TÜBITAK/GSU Summer School on Algebraic Geometry and Number Theory in Istanbul in 2014.

Mathematics Subject Classification (2010). Primary: 14C05. Secondary: 14E30, 14J60, 13D02.