Kyoto Journal of Mathematics

On a theorem of Castelnuovo and applications to moduli

Abel Castorena and Ciro Ciliberto

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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 g22, the answer is 2g+1, and it is attained by trigonal canonical curves varying on a balanced rational normal scroll.

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

First available in Project Euclid: 1 August 2011

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Zentralblatt MATH identifier

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


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.

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