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 of M ¯ g ′ \overline {\mathcal {M}}_{g’} by giving a complete enumerative description of the rational map between two moduli spaces of curves M ¯ g ⇢ M ¯ g ′ \overline {\mathcal {M}}_g \dashrightarrow \overline {\mathcal {M}}_{g’} which associates to a curve C C of genus g g its 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 on M ¯ g \overline {\mathcal {M}}_g of small slope, which in turn can be used to show that various moduli space of curves with level structure are of general type. For low g ′ g’ our calculation can be used to study the intersection theory of the moduli space of Prym varieties of dimension 5 5 .