Rational maps between moduli spaces of curves and Gieseker-Petri divisors

Abstract

We study contractions of the moduli space of stable curves beyond the minimal model ofM¯g′\overline {\mathcal {M}}_{g’}by giving a complete enumerative description of the rational map between two moduli spaces of curvesM¯g⇢M¯g′\overline {\mathcal {M}}_g \dashrightarrow \overline {\mathcal {M}}_{g’}which associates to a curveCCof genusggits Brill–Noether locus of special divisors in the case this locus is a curve. As an application we construct many examples of moving effective divisors onM¯g\overline {\mathcal {M}}_gof small slope, which in turn can be used to show that various moduli space of curves with level structure are of general type. For lowg′g’our calculation can be used to study the intersection theory of the moduli space of Prym varieties of dimension55.