The moduli space of curves and Gromov-Witten theory

Abstract

These notes are intended to explain how Gromov-Witten theory has been useful in understanding the moduli space of complex curves. We will focus on the moduli space of smooth curves and how much of the recent progress in understanding it has come through "enumerative" invariants in Gromov-Witten theory, something which we take for granted these days, but which should really be seen as surprising. There is one sense in which it should not be surprising - in many circumstances, modern arguments can be loosely interpreted as the fact that we can understand curves in general by studying branched covers of the complex projective line, as all curves can be so expressed. We will see this theme throughout the notes, from a Riemannstyle parameter count in Sect. 2.2 to the tool of relative virtual localization in Gromov-Witten theory in Sect. 5. These notes culminate in an approach to Faber's intersection number conjecture using relative Gromov-Witten theory (joint work with Goulden and Jackson [GJV3]). One motivation for this article is to convince the reader that our approach is natural and straightforward. © 2008 Springer-Verlag Berlin Heidelberg.