ON THE SLOPES OF THE MODULI SPACES OF CURVES

Abstract

Let Mg be the moduli space of smooth curves of genus g and Mg the moduli space of stable curves of genus g. Then Mg = MgU A, where A = YJg£ ] A*> Ao is the closure of the locus of genus g — 1 curves with one double point, and A* (i ^ 0) is the the closure of the locus of the stable curves of type (i,g — i). Let A be the class of the Hodge line bundle on A4S, let Si be the class of Aj if i ^ 1, and let Si be half of the class of Ai . Let S = So H h 4 [11], and on the moduli functor, the canonical class is 13A — 2S, the only difference is the coefficient of Si (see [11, 9] for the details). This class is called effective if n(aX — bS) is effective for sufficiently large and sufficiently divisible n. Harris and Morrison [10] define the slope of the moduli space as