  We carry out a detailed intersection theoretic analysis of the
Deligne-Mumford compactification of the divisor on M_{10} consisting of curves
sitting on K3 surfaces. This divisor is not of classical Brill-Noether type,
and is known to give a counterexample to the Slope Conjecture. The computation
relies on the fact that this divisor has four different incarnations as a
geometric subvariety of the moduli space of curves, one of them as a higher
rank Brill-Noether divisor consisting of curves with an exceptional rank 2
vector bundle. As an application we describe the birational nature of the
moduli space of n-pointed curves of genus 10, for all n. We also show that on
M_{11} there are effective divisors of minimal slope and having large Iitaka
dimension. This seems to contradict the belief that on M_g the classical
Brill-Noether divisors are essentially the only ones of slope 6+12/(g+1).
