Proceedings of the Second ISAAC Congress pp 971-983 | Cite as

# Variety of Special Nets of Degree g-1 on Double Coverings of a Smooth Plane Quartic of Genus 9

## Abstract

Let *C* be a complete non-singular curve over ℂ of genus *g*. We denote by *W* _{ d } ^{ r }(*C*) the subscheme of the Picard variety Pic^{ d }(C) whose support is the locus of complete linear series of degree *d* and dimension at least *r*. In case *d* > *g* + *r* - 1, *W* _{ d } ^{ r } (*C*) = Pic^{ d }(*C*) and if *d* = *g* + *r* - 1, *W* _{ d } ^{ r } (*C*) has dimension *d*. Therefore the dimension of *W* _{ d } ^{ r } (*C*) is independent of *C* in the range *d* ≥ *g* + *r* - 1. If *d* ≤ *g* + *r* - 2, one knows that dim*W* _{ d } ^{ r } (*C*) ≥ *p*(*d*, *g*, *r*):= *g* - (*r* + 1) (*g* - *d* + *r*) for any curve *C* and is equal to *p*(*d*, *g*, *r*) for general curve *C* (see Kleiman-Laksov [9] and Griffiths-Harris [6]). But the dimension of *W* _{ d } ^{ r }(*C*) might be greater than *p*(*d*, *g*, *r*) for some special curve *C*. Moreover, for curves *C* with dim *W* _{ d } ^{ r } (*C*) > *p*(*d*, *g*, *r*), *C* must be of some special type of curves. The first important result along this line is the following well-known theorem of H. Martens which has been extended by a theorem of D. Mumford.

### Keywords

Kato## Preview

Unable to display preview. Download preview PDF.

### References

- [1]E.Arbarello, M.Cornalba, P.A.Griffiths and J.Harris: Geometry of Algebraic curves I,
*Springer-Verlag*, 1985.MATHGoogle Scholar - [2]E. Ballico, C. Keem, G. Martens and A. Ohbuchi: On curves of genus eight,
*Math. Z.***227**, pp. 543–554, (1998)MathSciNetMATHCrossRefGoogle Scholar - [3]Kyung-Hye Cho, Changho Keem and Akira Ohbuchi: On the variety of special linear systems of degree
*g - 1*,*preprint*Google Scholar - [4]M. Coppens: Some remarks on the scheme W,,
*Ann. di Mat. puna ed applicata (4)*,**157**, pp. 183–197, (1990)MathSciNetMATHCrossRefGoogle Scholar - [5]M. Coppens, C. Keem and G. Martens: Primitive linear series on curves,
*Manuscripta Mathematica*,**77**, pp. 237–264, (1992)MathSciNetMATHCrossRefGoogle Scholar - [6]Griffiths and Harris: The dimension of the variety of special linear systems on a general curve,
*Duke Math. J.*,**47**, pp. 233–272, (1980)MathSciNetCrossRefGoogle Scholar - [7]C. Keem: On the variety of special linear systems on an algebraic curve,
*Math. Ann.*,**288**, pp. 309–322, (1990)MathSciNetMATHCrossRefGoogle Scholar - [8]T. Kato and A. Ohbuchi: Very ampleness of multiple of tetragonal linear systems,
*Comm. in Algebra*,**21**, pp.4587–4597, (1993)MathSciNetMATHCrossRefGoogle Scholar - [9]S. Kleiman and D. Laksov: On the existence of special divisors, Am.
*J. Math.*,**94**, pp. 431–436, (1972)MathSciNetMATHCrossRefGoogle Scholar - [10]G. Martens: On dimension theorems of the varieties of special divisors on a curve,
*Math. Ann.*,**267**, pp. 279–288, (1984)MathSciNetMATHCrossRefGoogle Scholar - [11]H. Martens: On the varieties of special divisors on a curve,
*J. Reine Angew. Math.*, 227, pp. 111–120, (1967)MathSciNetMATHCrossRefGoogle Scholar