EFFECTIVE DIVISORS ON Mg, CURVES ON K3 SURFACES, AND THE SLOPE CONJECTURE

In this paper we use the geometry of curves lying on K3 surfaces in order to obtain a number of results about effective divisors on the moduli space of stable curves Mg. We begin by showing two statements on the slopes of such divisors: first that the HarrisMorrison Slope Conjecture fails to hold on M10 and second, that in order to compute the slope of Mg for g ≤ 23, one only has to look at the coefficients of the classes λ and δ0 in the standard expansion in terms of the generators of the Picard group. The proofs are based on a general result providing inequalities between the first few coefficients of effective divisors on Mg. We then describe in detail the divisor K on M10 consisting of smooth sections of K3 surfaces, and its compactification K in the moduli space M10 (which is the counterexample to the conjecture mentioned above). As far as we know this is the first intersection theoretic analysis of a geometric subvariety on Mg which is not of classical Brill-Noether-Petri type, that is, a locus of curves carrying an exceptional linear series gd. Along the way, various other results on the Kodaira-Iitaka dimension of distinguished linear series on certain moduli spaces of (pointed) stable curves are obtained. We give the technical statements in what follows. 1