Effective divisors on M̄_g, curves on K3 surfaces, and the slope conjecture

We compute the class of the compactification of the divisor of curves sitting on a K3 surface and show that it violates the Harris-Morrison Slope Conjecture. We carry this out using the fact that this divisor has four distinct incarnations as a geometric subvariety of the moduli space of curves. We also give a counterexample to a hypothesis raised by Harris and Morrison that the Brill-Noether divisors are essentially the only effective divisors on the moduli space of curves having minimal slope 6 + 12/(g + 1).