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Gonality and Clifford index of projective curves on ruled surfaces

Authors: Youngook Choi and Seonja Kim
Journal: Proc. Amer. Math. Soc. 140 (2012), 393-402
MSC (2010): Primary 14H51, 14J26, 14H45
Published electronically: June 1, 2011
MathSciNet review: 2846309
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Abstract: Let $ X$ be a smooth curve on a ruled surface $ \pi : S\rightarrow C$. In this paper, we deal with the questions on the gonality and the Clifford index of $ X$ and on the composedness of line bundles on $ X$ with the covering morphism $ \pi \vert _X$. The main theorem shows that if a smooth curve $ X\sim aC_o +{\bf b}f$ satisfies some conditions on the degree of $ \bf b$, then a line bundle $ \mathcal{L}$ on $ X$ with $ \mathrm{Cliff}(\mathcal{L})\le ag(C)-1$ is composed with $ \pi \vert _X$. This implies that a part of the gonality sequence of $ X$ is computed by the gonality sequence of $ C$ as follows:

$\displaystyle d_r (X)=ad_r (C) ~~$ for $\displaystyle ~r\le L,$

where $ L$ is the length of the gonality sequence of $ C$.

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Additional Information

Youngook Choi
Affiliation: Department of Mathematics Education, Yeungnam University, 214-1 Daedong Gyeongsan, 712-749, Gyeongsangbuk-do, Republic of Korea

Seonja Kim
Affiliation: Department of Electronics, Chungwoon University, Hongseong, Chungnam, 350-701, Republic of Korea

Keywords: Gonality, Clifford index, ruled surface, multiple covering, Castelnuovo-Severi inequality, gonality sequence.
Received by editor(s): September 28, 2009
Received by editor(s) in revised form: November 16, 2010
Published electronically: June 1, 2011
Additional Notes: The first author’s work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-314-C00011)
The second author’s work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (2009-0075469)
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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