Relations among divisors on themoduli space of curveswithmarked points

Let Mg be the coarse moduli space of stable curves of genus g. Eisenbud, Harris and Mumford proved a relation between certain divisors on Mg (the Brill-Noether divisors, to be described below). Calculating their classes in PicMg⊗Q, they succeeded in proving that Mg is of general type for g > 23, g+1 composite. In subsequent work, the restriction that g + 1 be composite was removed. In this paper, I will generalize their relation to Mg,n, the moduli space of stable curves of genus g with n marked points. This does not yield new results on the Kodaira dimension of Mg,n, as the “divisors of Brill-Noether type”, which I introduce below, are less effective for this purpose than certain other divisors which I studied in my dissertation [L]. (I expect to publish the other results of [L] shortly.) The remainder of this introduction will be devoted to stating the results. For basic facts on Mg and Mg,n, the reader is referred to [HM] and [K] respectively. Definition. Fix a nonnegative integer g, and let r > 1, d > 1 be integers such that g−(r+1)(g−d+r) = −1. (Here, the left side is the expected dimension of the space of g d’s on a curve of genus g.) A divisor of BrillNoether type on Mg is a codimension-1 component of the locus of curves which have an admissible g r d. (All g d’s on nonsingular irreducible curves are admissible; in general, an admissible g r d is a g r d on each component with certain ramification conditions at the singular points of the curve. Again, see [HM] for details.) The Brill-Noether Ray Theorem of Eisenbud, Harris, and Mumford then asserts that, for fixed g, all of these divisors on Mg are linearly dependent, and calculates their class. We generalize their definition and theorem as follows: