The nonabelian Brill-Noether divisor on (M)over-tilde13 and the Kodaira dimension of (R)over-tilde13

We highlight several novel aspects of the moduli space of curves of genus 13, the first genus g where phenomena related to K3 surfaces no longer govern the birational geometry of (M) over bar (g). We compute the class of the nonabelian Brill-Noether divisor on (M) over bar (13) of curves that have a stable rank-two vector bundle with canonical determinant and many sections. This provides the first example of an effective divisor onMg with slope less than 6C10=g. Earlier work on the slope conjecture suggested that such divisors may not exist. The main geometric application of our result is a proof that the Prym moduli space (R) over bar (13) is of general type. Among other things, we also prove the Bertram-Feinberg-Mukai and the strong maximal rank conjectures on (M) over bar (13).