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 M₁₃ 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 R₁₃ is of general type. Among other things, we also prove the Bertram–Feinberg–Mukai and the strong maximal rank conjectures on M₁₃.