Curves of genus whose canonical model lies on a surface of degree
Author:
Gianfranco Casnati
Journal:
Proc. Amer. Math. Soc. 141 (2013), 437450
MSC (2010):
Primary 14N25; Secondary 14H51, 14H30, 14N05
Published electronically:
June 12, 2012
MathSciNet review:
2996948
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Similar Articles 
Additional Information
Abstract: Let be a nonhyperelliptic curve of genus . We prove that if the minimal degree of a surface containing the canonical model of in is , then either and carries exactly one or and is birationally isomorphic to a plane septic curve with at most double points as singularities.
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plane models”, Manuscripta Math. 71 (1991),
no. 3, 337–338. MR 1103738
(92a:14027b), 10.1007/BF02568410
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27 (2004), no. 1, 137–147. MR 2060080
(2005d:14046), 10.3836/tjm/1244208480
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SaintDonat, On Petri’s analysis of the linear system of
quadrics through a canonical curve, Math. Ann. 206
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Oscar
Zariski, A simplified proof for the resolution of singularities of
an algebraic surface, Ann. of Math. (2) 43 (1942),
583–593. MR 0006851
(4,52c)
 [ACGH]
 E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, Geometry of algebraic curves, vol. I, Springer, 1985. MR 770932 (86h:14019)
 [BCF]
 E. Ballico, G. Casnati, C. Fontanari, On the geometry of bihyperelliptic curves,
J. Korean Math. Soc. 44 (2007), 13391350. MR 2358958 (2008i:14040)
 [BCN]
 E. Ballico, G. Casnati, R. Notari, Canonical curves with low apolarity, J. Algebra 332 (2011), 229243. MR 2774686
 [Br]
 J. Brawner, Tetragonal curves, scrolls and surfaces, Trans. Amer. Math. Soc. 349 (1997), 30753091. MR 1401515 (97j:14056)
 [Cs]
 G. Casnati, Canonical curves on surfaces of very low degree, Proc. Amer. Math. Soc. 140 (2012), 11851197.
 [CE]
 G. Casnati, T. Ekedahl, Covers of algebraic varieties I. A general structure theorem, covers of degree , and Enriques surfaces, J. Algebraic Geom. 5 (1996), 439460. MR 1382731 (97c:14014)
 [Ch]
 G. Chaves, Revêtement ramifiés de la droite projective complexe, Math. Z. 226 (1997), 6784. MR 1472141 (98j:14038)
 [CH]
 C. Ciliberto, J. Harris, Surfaces of low degree containing a general canonical curve, Comm. Algebra 27 (1999), 11271140. MR 1669124 (2000c:14051)
 [CK1]
 M. Coppens, T. Kato, The gonality of smooth curves with plane models, Manuscripta Math. 70 (1990), 525. MR 1080899 (92a:14027a)
 [CK2]
 M. Coppens, T. Kato, Correction to the gonality of smooth curves with plane models, Manuscripta Math. 71 (1991), 337338. MR 1103738 (92a:14027b)
 [Co]
 I. Coşkun, Surfaces of low degree containing a canonical curve. To appear in Contemp. Math.
 [De]
 M. Demazure, Surfaces de Del Pezzo  II, III, IV, V, Séminaire sur les singularités des surfaces, Palaiseau, France 19761977 (M. Demazure, H. Pinkham, B. Teissier, eds.), Lecture Notes in Math. 777, Springer, 1980.
 [Ha]
 R. Hartshorne, Algebraic geometry, Springer, 1977. MR 0463157 (57:3116)
 [Mi]
 J.C. Migliore, Introduction to liaison theory and deficiency modules, Progress in Mathematics, vol. 165, Birkhäuser, 1998. MR 1712469 (2000g:14058)
 [OS]
 M. Ohkouchi, F. Sakai, The gonality of singular plane curves, Tokyo J. Math. 27 (2004), 137147. MR 2060080 (2005d:14046)
 [SD]
 B. SaintDonat, On Petri's analysis of the linear system of quadrics through a canonical curve, Math. Ann. 206 (1973), 157175. MR 0337983 (49:2752)
 [Sch1]
 F.O. Schreyer, Syzygies of canonical curves and special linear series, Math. Ann. 275 (1986), 105137. MR 849058 (87j:14052)
 [Sch2]
 F.O. Schreyer, A standard basis approach to syzygies of canonical curves, J. Reine Angew. Math. 421 (1991), 83123. MR 1129577 (92j:14040)
 [Za]
 O. Zariski, A simplified proof for the resolution of singularities of an algebraic surface, Ann. of Math. (2) 43 (1942), 583593. MR 0006851 (4:52c)
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Additional Information
Gianfranco Casnati
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, c. so Duca degli Abruzzi 24, 10129 Torino, Italy
Email:
casnati@calvino.polito.it
DOI:
http://dx.doi.org/10.1090/S000299392012113358
Keywords:
Curve,
canonical model,
tetragonality.
Received by editor(s):
March 21, 2011
Received by editor(s) in revised form:
July 1, 2011
Published electronically:
June 12, 2012
Additional Notes:
This work was done in the framework of PRIN ‘Geometria delle varietà algebriche e dei loro spazi di moduli’, cofinanced by MIUR (COFIN 2008)
Communicated by:
Lev Borisov
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
