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

Published electronically:
July 29, 2011

MathSciNet review:
2869104

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a non-hyperelliptic curve of genus . We recall some facts about curves endowed with a base-point-free . Then we prove that if the minimal degree of a surface containing the canonical model of in is , then and carries exactly one . As a by-product, we deduce that if the canonical model of in is contained in a surface of degree at most , then is either trigonal or tetragonal or isomorphic to a plane sextic.

**[Ab]**Shreeram Abhyankar,*Local uniformization on algebraic surfaces over ground fields of characteristic 𝑝≠0*, Ann. of Math. (2)**63**(1956), 491–526. MR**0078017****[A-C-G-H]**E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris,*Geometry of algebraic curves. Vol. I*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR**770932****[B-C-N]**E. Ballico, G. Casnati, R. Notari,*Canonical curves with low apolarity*, J. Algebra,**332**(2011), 229-243.**[Bd-Sc]**Markus Brodmann and Peter Schenzel,*Arithmetic properties of projective varieties of almost minimal degree*, J. Algebraic Geom.**16**(2007), no. 2, 347–400. MR**2274517**, 10.1090/S1056-3911-06-00442-5**[Bn-Sa]**M. Brundu, G. Sacchiero,*Stratification of the moduli space of four-gonal curves*, preprint.**[Cs]**G. Casnati,*Covers of algebraic varieties. III. The discriminant of a cover of degree 4 and the trigonal construction*, Trans. Amer. Math. Soc.**350**(1998), no. 4, 1359–1378. MR**1467462**, 10.1090/S0002-9947-98-02136-9**[C-E]**G. Casnati and T. Ekedahl,*Covers of algebraic varieties. I. A general structure theorem, covers of degree 3,4 and Enriques surfaces*, J. Algebraic Geom.**5**(1996), no. 3, 439–460. MR**1382731****[C-H]**C. Ciliberto and J. Harris,*Surfaces of low degree containing a general canonical curve*, Comm. Algebra**27**(1999), no. 3, 1127–1140. MR**1669124**, 10.1080/00927879908826485**[dP1]**P. del Pezzo,*Sulle superficie di ordine nello spazio di dimensioni*, Rend. R. Acc. delle Scienze Fisiche e Mat. di Napoli,**24**(1885), 212-216.**[dP2]**P. del Pezzo,*Sulle superficie di ordine nello spazio di dimensioni*, Rend. del Circolo Mat. di Palermo,**1**(1886), 241-271.**[DP-Z1]**P. De Poi, F. Zucconi,*Gonality, apolarity, and hypercubics*, J. London Math. Soc., to appear.**[DP-Z2]**P. De Poi, F. Zucconi,*Fermat hypersurfaces and subcanonical curves*, arXiv:0908.0522.**[E-H]**David Eisenbud and Joe Harris,*On varieties of minimal degree (a centennial account)*, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 3–13. MR**927946**, 10.1090/pspum/046.1/927946**[Fj1]**Takao Fujita,*Classification of projective varieties of Δ-genus one*, Proc. Japan Acad. Ser. A Math. Sci.**58**(1982), no. 3, 113–116. MR**664549****[Fj2]**Takao Fujita,*Projective varieties of Δ-genus one*, Algebraic and topological theories (Kinosaki, 1984) Kinokuniya, Tokyo, 1986, pp. 149–175. MR**1102257****[Fj3]**Takao Fujita,*Classification theories of polarized varieties*, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990. MR**1162108****[Ha]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[H-W]**Fumio Hidaka and Keiichi Watanabe,*Normal Gorenstein surfaces with ample anti-canonical divisor*, Tokyo J. Math.**4**(1981), no. 2, 319–330. MR**646042**, 10.3836/tjm/1270215157**[I-K]**Anthony Iarrobino and Vassil Kanev,*Power sums, Gorenstein algebras, and determinantal loci*, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, Berlin, 1999. Appendix C by Iarrobino and Steven L. Kleiman. MR**1735271****[I-R]**Atanas Iliev and Kristian Ranestad,*Canonical curves and varieties of sums of powers of cubic polynomials*, J. Algebra**246**(2001), no. 1, 385–393. MR**1872627**, 10.1006/jabr.2001.8942**[Mi]**Juan C. Migliore,*Introduction to liaison theory and deficiency modules*, Progress in Mathematics, vol. 165, Birkhäuser Boston, Inc., Boston, MA, 1998. MR**1712469****[R-S]**Kristian Ranestad and Frank-Olaf Schreyer,*Varieties of sums of powers*, J. Reine Angew. Math.**525**(2000), 147–181. MR**1780430**, 10.1515/crll.2000.064**[SD]**B. Saint-Donat,*On Petri’s analysis of the linear system of quadrics through a canonical curve*, Math. Ann.**206**(1973), 157–175. MR**0337983****[Sch1]**Frank-Olaf Schreyer,*Syzygies of canonical curves and special linear series*, Math. Ann.**275**(1986), no. 1, 105–137. MR**849058**, 10.1007/BF01458587**[Sch2]**Frank-Olaf Schreyer,*A standard basis approach to syzygies of canonical curves*, J. Reine Angew. Math.**421**(1991), 83–123. MR**1129577**, 10.1515/crll.1991.421.83**[Za]**Oscar Zariski,*A simplified proof for the resolution of singularities of an algebraic surface*, Ann. of Math. (2)**43**(1942), 583–593. MR**0006851**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
14N25,
14H51,
14H30,
14N05

Retrieve articles in all journals with MSC (2010): 14N25, 14H51, 14H30, 14N05

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:
http://dx.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.