Weierstrass points of the universal curve

Lax, R. F.

doi:10.1007/bf02547970

Cited by 25 publications

(28 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The map d % will be said, in the following, by a slight abuse of terminology, the exterior derivative along the fibers of % (compare with [23], where in the case of families of smooth curves, an analytic description of d % is provided). As it is well known, 3 % and d % enjoy some nice functorial properties.…”

Section: 3mentioning

confidence: 99%

“…n !, of a vector bundle which (following Lax, [23]) shall be denoted as J n % l, and which is, by definition, the n-th jets bundle of l along the fibers of %. The above outlined construction is the relative version of the jet bundles for Gorenstein curves constructed in [12].…”

Section: 4mentioning

confidence: 99%

“…The above outlined construction is the relative version of the jet bundles for Gorenstein curves constructed in [12]. If % X X À3 S is a family of smooth curves and l 3 % is the relative canonical sheaf, then J n 3 % is exactly the n-th jet of the relative canonical bundle along the fibers defined by Lax in [23]. There, the construction was performed by using Patt's local coordinates ( [28]) in the Teichmu« ller space.…”

Section: 4mentioning

confidence: 99%

See 2 more Smart Citations

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

Gatto¹

2001

MATH. SCAND.

View full text Add to dashboard Cite

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 ...

show abstract

Section: 3mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

See 1 more Smart Citation

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

Gatto¹

2001

MATH. SCAND.

View full text Add to dashboard Cite

show abstract

“…and let π: V->T g denote the universal curve of genus g. Let Wζ, for k -2, , 2g -2 and r -1, 2, , denote the closed complex subspaces of V of Weierstrass points of the universal curve which were defined in [1], [2]. These spaces may be described set-theoretically as follows: 1) for k £ g, then I 2^ϊΊ = {(*, P) £ V: in the Weierstrass gap sequence at Pe V t , there are at least r nongaps ^&}.…”

Section: * Let T G Denote the Teichmuller Space For Teichmuller Surfmentioning

confidence: 99%

“…Then £ is determined by its values on a set of local parameters of <^V, (ί , P ). So, if f(ί) = we, ξ(c m ) = 6 m ε, m = 1,,then (u,b u , 6 3ff -3 ) serve as coordinates for T (ί , P) (F) (V).…”

Section: Definition Put W~{γ) = {(T P) E V: γ{P) = γ)mentioning

confidence: 99%