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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Derivatives of Wronskians with applications
to families of special Weierstrass points


Authors: Letterio Gatto and Fabrizio Ponza
Journal: Trans. Amer. Math. Soc. 351 (1999), 2233-2255
MSC (1991): Primary 14H10, 14H15, 14H55, 14H99
Published electronically: February 4, 1999
MathSciNet review: 1615963
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Abstract: Let $\pi :\mathfrak{X}\longrightarrow S$ be a flat proper family of smooth connected projective curves parametrized by some smooth scheme of finite type over $\mathbb{C}$. On every such a family, suitable derivatives along the fibers" (in the sense of Lax) of the relative wronskian, as defined by Laksov and Thorup, are constructed. They are sections of suitable jets extensions of the $g(g+1)/2$-th tensor power of the relative canonical bundle of the family itself.

The geometrical meaning of such sections is discussed: the zero schemes of the $(k-1)$-th derivative ($k\geq 1$) of a relative wronskian correspond to families of Weierstrass Points (WP's) having weight at least $k$.

The locus in $M_{g}$, the coarse moduli space of smooth projective curves of genus $g$, of curves possessing a WP of weight at least $k$, is denoted by $wt(k)$. The fact that $wt(2)$ has the expected dimension for all $g\geq 2$ was implicitly known in the literature. The main result of this paper hence consists in showing that $wt(3)$ has the expected dimension for all $g\geq 4$. As an application we compute the codimension $2$ Chow ($Q$-)class of $wt(3)$ for all $g\geq 4$, the main ingredient being the definition of the $k$-th derivative of a relative wronskian, which is the crucial tool which the paper is built on. In the concluding remarks we show how this result may be used to get relations among some codimension $2$ Chow ($Q$-)classes in $M_{4}$ ($g\geq 4$), corresponding to varieties of curves having a point $P$ with a suitable prescribed Weierstrass Gap Sequence, relating to previous work of Lax.


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Additional Information

Letterio Gatto
Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24-10129 Torino, Italy
Email: lgatto@polito.it

Fabrizio Ponza
Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24-10129 Torino, Italy
Email: ponza@dm.unito.it

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02343-0
PII: S 0002-9947(99)02343-0
Keywords: Relative wronskians, derivatives of relative wronskians, families of Weierstrass points, moduli spaces of curves, Chow classes in $M_{g}$
Received by editor(s): February 5, 1997
Published electronically: February 4, 1999
Additional Notes: Work partially supported by GNSAGA-CNR, MURST and by Dottorato di Ricerca in Matematica, Consorzio Universitario Torino-Genova
Article copyright: © Copyright 1999 American Mathematical Society