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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
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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  • [Ab] S. S. Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic $ p\ne 0$, Ann. of Math. (2), 63 (1956), 491-526. MR 0078017 (17:1134d)
  • [A-C-G-H] E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of algebraic curves, Vol. I, Springer-Verlag, 1985. MR 0770932 (86h:14019)
  • [B-C-N] E. Ballico, G. Casnati, R. Notari, Canonical curves with low apolarity, J. Algebra, 332 (2011), 229-243.
  • [Bd-Sc] M. Brodmann, P. Schenzel, Arithmetic properties of projective varieties of almost minimal degree, J. Algebraic Geom., 16 (2007), 347-400. MR 2274517 (2008b:14085)
  • [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), 1359-1378. MR 1467462 (98i:14021)
  • [C-E] G. Casnati, T. Ekedahl, Covers of algebraic varieties I. A general structure theorem, covers of degree $ 3$, $ 4$ and Enriques surfaces, J. Algebraic Geom., 5 (1996), 439-460. MR 1382731 (97c:14014)
  • [C-H] C. Ciliberto, J. Harris, Surfaces of low degree containing a general canonical curve, Comm. Algebra, 27 (1999), 1127-1140. MR 1669124 (2000c:14051)
  • [dP1] P. del Pezzo, Sulle superficie di ordine $ n$ nello spazio di $ n+1$ dimensioni, Rend. R. Acc. delle Scienze Fisiche e Mat. di Napoli, 24 (1885), 212-216.
  • [dP2] P. del Pezzo, Sulle superficie di ordine $ n$ nello spazio di $ n$ 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] D. Eisenbud, J. Harris, On varieties of minimal degree (a centennial account), Algebraic Geometry, Bowdoin, 1985, Spencer J. Bloch, Proceedings of Symposia in Pure Mathematics, 46, 3-13, AMS, 1987. MR 927946 (89f:14042)
  • [Fj1] T. Fujita, Classification of projective varieties of $ \Delta$-genus one, Proc. Japan Acad. - Ser. A, 58 (1982), 113-116. MR 664549 (83g:14003)
  • [Fj2] T. Fujita, Projective varieties of $ \Delta$-genus one, Algebraic and topological theories. Papers from the symposium dedicated to the memory of Dr. Takehiko Miyata (Kinokuniya, Tokyo, 1985), M. Nagata, S. Araki, A. Hattori, N. Iwahori et al. (eds.), Kinokuniya Company Ltd., 1986, 149-175. MR 1102257
  • [Fj3] T. Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Notes Series, 155, Cambridge University Press, 1990. MR 1162108 (93e:14009)
  • [Ha] R. Hartshorne, Algebraic geometry, Springer, 1977. MR 0463157 (57:3116)
  • [H-W] F. Hidaka, K. Watanabe, Normal Gorenstein surfaces with ample anti-canonical divisor, Tokyo J. Math., 4 (1981), 319-330. MR 646042 (83h:14031)
  • [I-K] A. Iarrobino, V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, 1721, Springer, 1999. MR 1735271 (2001d:14056)
  • [I-R] A. Iliev, K. Ranestad, Canonical curves and varieties of sums of powers of cubic polynomials, J. Algebra, 246 (2001), 385-393. MR 1872627 (2003c:14032)
  • [Mi] J. C. Migliore, Introduction to liaison theory and deficiency modules, Progress in Mathematics, 165, Birkhäuser, 1998. MR 1712469 (2000g:14058)
  • [R-S] K. Ranestad, F. O. Schreyer, Varieties of sums of powers, J. Reine Angew. Math., 525 (2000), 147-181. MR 1780430 (2001m:14009)
  • [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 (49:2752)
  • [Sch1] F. O. Schreyer, Syzygies of canonical curves and special linear series, Math. Ann., 275 (1986), 105-137. MR 849058 (87j:14052)
  • [Sch2] F. O. Schreyer, A standard basis approach to syzygies of canonical curves, J. Reine Angew. Math., 421 (1991), 83-123. 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), 583-593. MR 0006851 (4:52c)

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Additional Information

G. Casnati
Affiliation: Dipartimento di Matematica, Politecnico di Torino, Duca degli Abruzzi 24, 10129 Torino, Italy

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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