Koszul divisors on moduli spaces of curves

Given a moduli space, how can one construct the ``best'' (in the sense of higher dimensional algebraic geometry) effective divisor on it? We show that, at least in the case of the moduli space of curves, the answer is provided by the Koszul divisor defined in terms of the syzygies of the parameterized objects. In this paper, we find a formula for the slopes of all Koszul divisors on $\overline{{\cal M}}_g$. In particular, we obtain the first infinite series of counterexamples to the Harris-Morrison Slope Conjecture and we prove the Maximal Rank Conjecture in the case when the Brill-Noether number of the corresponding linear series equals~$0$. We also find shorter proofs for the formulas of the class of the Brill-Noether and Gieseker-Petri divisors. Finally, we improve most of Logan's results on the Kodaira dimension of the moduli spaces $\overline{{\cal M}}_{g, n}$ of pointed stable curves.