Let wt2 be the closure in M g , the coarse moduli space of stable complex curves of genus g ! 3, of the locus in M g of curves possessing a Weierstrass point of weight at least 2. The class of wt2 in the group PicM g Q is computed. The computation heavily relies on the notion of`d erivative'' of a relative Wronskian, introduced in [15] for families of smooth curves and here extended to suitable families of Deligne-Mumford stable curves. Such a computation provides, as a byproduct, a simpler proof of the main result proven in [6].0. Introduction 0.1. Let M g (resp. M g ) be the coarse moduli space of complex smooth (resp. Deligne-Mumford stable) projective curves of genus g ! 3 and let wt2 be the subset of M g defined by the locus of (isomorphism classes of) curves possessing a Weierstrass point of weight at least 2. Such a set has been equipped with a scheme structure by Ponza in his doctoral thesis ([29]; see also [15]) by using the notion of derivative of the wronskian relative to a proper f lat family of smooth curves. The locus wt2 turns out to be a divisor in M g , and the purpose of this paper is to compute the class of wt2, its closure in M g , in the group PicM g Q. The result is gotten by extending the notion of derivative of the relative wronskian (see sect. 2 for details) to a family of stable curves whose general fiber is smooth and non-hyperelliptic, so providing a new application of the tools introduced in [15].
0.2.As one may reasonably expect, the divisor wt2 is strongly related to two other natural divisors, defined in terms of curves possessing some special Weierstrass points, which have been extensively studied in the literature. The first one is the locus D gÀ1 of the curves having a Weierstrass point whose first non gap is g À 1. The second one is the locus E1 of the curves possessing a Weierstrass point of type g 1: a point P of a curve C of genus MATH. SCAND. 88 (2001)
, 41^71Ã Work sponsored by a CNR-NATO research fellowship and, partially, by GNSAGA-CNR and MURST. Received May 12, 1998. g is said to be of type g 1 if there exists a non zero canonical divisor containing nP, with n ! g 1. Let us denote by D gÀ1 and by E1 the closures, respectively, of D gÀ1 and E1 in M g . The class of D gÀ1 and E1 in PicM g Q have been computed respectively by Diaz ([7]) and Cukierman ([6]).Both computations are based on the theory of the compactification of the Hurwitz scheme by means of the admissible covers, according to Harris and Mumford ([21]). 0.3. Roughly speaking, Diaz gets his results by an enumeration of all the possible admissible coverings which may occur as a degeneration of families of curves whose general fiber has a Weierstrass point whose first non gap is g À 1. Conceptually, this amounts to consider``curves'' in the boundary of M g and to compute their intersections with the divisor D gÀ1 .Cukierman's computations involve, instead, a fine analysis of the singularities of the closure of the Weierstrass locus w, that sits in the``universal curve'' over M g , along the locus ...
