Kyoto Journal of Mathematics

On a theorem of Castelnuovo and applications to moduli

Abstract

In this paper we prove a theorem stated by Castelnuovo which bounds the dimension of linear systems of plane curves in terms of two invariants, one of which is the genus of the curves in the system. This extends a previous result of Castelnuovo and Enriques. We classify linear systems whose dimension belongs to certain intervals which naturally arise from Castelnuovo’s theorem. Then we make an application to the following moduli problem: what is the maximum number of moduli of curves of geometric genus $g$ varying in a linear system on a surface? It turns out that, for $g\ge 22$, the answer is $2g+1$, and it is attained by trigonal canonical curves varying on a balanced rational normal scroll.

Article information

Source
Kyoto J. Math., Volume 51, Number 3 (2011), 633-645.

Dates
First available in Project Euclid: 1 August 2011

https://projecteuclid.org/euclid.kjm/1312205241

Digital Object Identifier
doi:10.1215/21562261-1299909

Mathematical Reviews number (MathSciNet)
MR2824002

Zentralblatt MATH identifier
1226.14047

Subjects
Primary: 14C20: Divisors, linear systems, invertible sheaves
Secondary: 14J26: Rational and ruled surfaces

Citation

Castorena, Abel; Ciliberto, Ciro. On a theorem of Castelnuovo and applications to moduli. Kyoto J. Math. 51 (2011), no. 3, 633--645. doi:10.1215/21562261-1299909. https://projecteuclid.org/euclid.kjm/1312205241

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