  We perform an intersection theoretic study of the rational map between two
different moduli spaces of stable curves which associates to a curve its
corresponding Brill-Noether locus (in the case this locus has virtual dimension
1). We then use these results to describe the cone of moving divisors on M_g.
Several other applications to moduli spaces of Prym varieties are presented. In
a different direction, we prove that the locus in M_g of curves failing to
satisfy the Gieseker-Petri theorem is supported in codimension 1 for every
possible type of linear series.
