The global geometry of the moduli space of curves

This is a survey written for the Proceedings of the AMS Summer Institute in Algebraic Geometry held in Seattle in 2005. Topics discussed in the survey include the ample and the effective cone of the moduli space of curves, Kodaira dimension, Slope Conjecture, log canonical models etc.Comment: 23 pages. Minor revisions. To appear in the Proceedings of the AMS Summer Research Institute in Algebraic Geometry-Seattle 200

Prym varieties and their moduli

We discuss the geometry of the moduli space of Prym varieties. The article is based on series of lectures given in Bedlewo and Luminy. The first section of the paper contains a detailed historical account of the lives of Friedrich Prym and Friedrich Schottky.Comment: 35 pages, minor corrections and additions. To appear in "Contributions to algebraic geometry" edited by P. Pragacz and published by the EM

The Geometry of the Moduli Space of Curves of Genus 23

We prove that the Kodaira dimension of the moduli space M_{23} of curves of genus 23 is at least 2. We also present some evidence for the hypothesis that the Kodaira dimension of the moduli space is actually equal to 2. Note that for g > 23 the moduli space is of general type, while for g\leq 22, Harris and Morrison conjectured that M_g is uniruled. The result on M_{23} is obtained by investigating the relative position of three explicit multicanonical divisors which are of Brill-Noether type.Comment: 21 pages, Late

Brill-Noether Loci and the Gonality Stratification of MgM_g

A Brill-Noether locus is a subvariety of M_g consisting of curves having certain linear series g^r_d. We study the relative position of Brill-Noether loci with respect to the gonality stratification of M_g. We construct smooth curves in P^r of given degree and genus and having the `expected' gonality. As an application we give a new proof of our result about the Kodaira dimension of the moduli space of curves of genus 23.Comment: 14 pages, latex. Revised version as it will appear in Crelles Journa