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Canonical curves on surfaces of very low degree


Author: G. Casnati
Journal: Proc. Amer. Math. Soc. 140 (2012), 1185-1197
MSC (2010): Primary 14N25; Secondary 14H51, 14H30, 14N05
DOI: https://doi.org/10.1090/S0002-9939-2011-10979-1
Published electronically: July 29, 2011
MathSciNet review: 2869104
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Abstract: Let $ C$ be a non-hyperelliptic curve of genus $ g$. We recall some facts about curves endowed with a base-point-free $ g^{1}_{4}$. Then we prove that if the minimal degree of a surface containing the canonical model of $ C$ in $ \check{\mathbb{P}}^{g-1}_k$ is $ g$, then $ 7\le g\le 12$ and $ C$ carries exactly one $ g^{1}_{4}$. As a by-product, we deduce that if the canonical model of $ C$ in $ \check{\mathbb{P}}^{g-1}_k$ is contained in a surface of degree at most $ g$, then $ C$ is either trigonal or tetragonal or isomorphic to a plane sextic.


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Additional Information

G. Casnati
Affiliation: Dipartimento di Matematica, Politecnico di Torino, c.so Duca degli Abruzzi 24, 10129 Torino, Italy
Email: casnati@calvino.polito.it

DOI: https://doi.org/10.1090/S0002-9939-2011-10979-1
Keywords: Curve, canonical model, tetragonality, Maroni number, apolarity.
Received by editor(s): October 14, 2010
Received by editor(s) in revised form: December 15, 2010, December 26, 2010, and December 29, 2010
Published electronically: July 29, 2011
Additional Notes: This work was done in the framework of PRIN ’Geometria delle varieté a algebriche e dei loro spazi di moduli’, cofinanced by MIUR (COFIN 2008)
Communicated by: Lev Borisov
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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