The nonabelian Brill–Noether divisor on ℳ13 and the Kodaira dimension of ℛ13

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 g .We compute the class of the nonabelian Brill-Noether divisor on M 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 on M g with slope less than 6 C 10=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 13 is of general type.Among other things, we also prove the Bertram-Feinberg-Mukai and the strong maximal rank conjectures on M 13 .14H10; 14T20 1. Introduction 803 2. The failure locus of the strong maximal rank conjecture on e M 13 809 3. The class of the virtual divisor e D 13 817 4. The strong maximal rank conjecture in genus 13 821 5. Effectivity of the virtual class 842 6.The Bertram-Feinberg-Mukai conjecture in genus 13 846 7. The nonabelian Brill-Noether divisor on M 13 853 8.The Kodaira dimension of R 13