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

DOI:
https://doi.org/10.1090/S0002-9947-99-02343-0

Published electronically:
February 4, 1999

MathSciNet review:
1615963

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a flat proper family of smooth connected projective curves parametrized by some smooth scheme of finite type over . 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 -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 -th derivative () of a relative wronskian correspond to families of Weierstrass Points (WP's) having weight at least .

The locus in , the coarse moduli space of smooth projective curves of genus , of curves possessing a WP of weight at least , is denoted by . The fact that has the expected dimension for all was implicitly known in the literature. The main result of this paper hence consists in showing that has the expected dimension for all . As an application we compute the codimension Chow (-)class of for all , the main ingredient being the definition of the -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 Chow (-)classes in (), corresponding to varieties of curves having a point 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:
https://doi.org/10.1090/S0002-9947-99-02343-0

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