## Canonical curves on surfaces of very low degree

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## Abstract:

Let $C$ be a nonâ€“hyperelliptic curve of genus $g$. We recall some facts about curves endowed with a baseâ€“pointâ€“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 byâ€“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.## References

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## Additional Information

**G. Casnati**- Affiliation: Dipartimento di Matematica, Politecnico di Torino, c.so Duca degli Abruzzi 24, 10129 Torino, Italy
- MR Author ID: 313798
- Email: casnati@calvino.polito.it
- 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 \lq Geometria delle varietĂ© a algebriche e dei loro spazi di moduli\rq, cofinanced by MIUR (COFIN 2008)
- Communicated by: Lev Borisov
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2869104