  Given a moduli problem with a nice coarse moduli space, which is the most
intrinsic and natural divisor on this moduli space? We describe a general
method of constructing effective divisors on a large class of moduli spaces
using the syzygies of the parametrized objects. In this paper we treat the case
of the moduli space of curves and we compute the slope of a doubly infinite
sequence of Koszul divisors on M_g. We present a single formula which apart
from giving myriads of new counterexamples to the Harris-Morrison Slope
Conjecture it also encodes virtually all interesting divisor class calculations
on M_g. In particular our formula specializes to earlier results due to
Harris-Mumford (the case of the Brill-Noether divisors), Eisenbud-Harris
(Giseker-Petri divisors), Khosla as well as to our earlier counterexamples to
the Slope Conjecture obtained using K3 surfaces and Hurwitz schemes. We also
describe five ways of constructing Koszul divisors on moduli spaces of pointed
curves. As an application, we improve most of Logan's results on the Kodaira
type of M_{g,n}.
