On the closure in $\overline M_g$ of smooth curves having a special Weierstrass point

Abstract

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 ...

Jets of line bundles on curves and Wronskians

“…If C were singular the right-hand side of (1) would not be locally free. But for locally complete intersection curves, one may use the locally free substitutes of the principal parts, constructed via a ''paste and glue'' procedure in [8,9], used in [12] and more elegantly described, in an intrinsic way, by Laksov and Thorup in [15].…”

Local Complete Intersections and Weierstrass Points

“…In this direction, there are two remarkable classical works by E. Arbarello, namely [Arba74,Arba78], where are studied the subloci W n,g ⊂ M g of curves admitting a Weierstrass point of weight not smaller than n. Among many others beautiful and high standing works describing cycles coming from Weierstrass points, for instance the remarkable [Di85], we would like to cite a particular one due to L. Gatto, c.f. [Ga01]. In this work, Gatto computes the class in the Picard group Pic(M g ) ⊗ Q of the locus of curves admitting a Weierstrass point of weight at least 2.…”

On the locus of curves with an odd subcanonical marked point

“…To do this, we make the substitutions X i → t i and solve a homogeneous linear system with 60 equations. We can solve it in way that the solution depends only on the 15 coefficients: Therefore the compactified moduli space M S 6,1 can be realized as a closed subset of the 14-dimensional weighted projective space P(T 1,− k[S]|k ) ∼ = P 14 α , where α = (2,3,4,4,5,5,6,6,7,8,8,9,10,11,12). Since the odd symmetric semigroup S is negatively graded, cf.…”

“…Stoehr's techniques avoid suitable classes of symmetric semigroups, more precisely, it is assumed that the multiplicity n 1 of S satisfies 3 < n 1 < g, and that S = 4, 5 , avoiding the general points of H hyp 2g−2 and of H odd 2g−2 of the Kontsevich-Zorich space H 2g−2 . Another successful approach to study families of Weierstrass points can be done by considering (generalized) Wronskians and its derivatives, we refer to [12,13].…”