Kyoto Journal of Mathematics

On a theorem of Castelnuovo and applications to moduli

Abel Castorena and Ciro Ciliberto

Full-text: Open access

Abstract

In this paper we prove a theorem stated by Castelnuovo which bounds the dimension of linear systems of plane curves in terms of two invariants, one of which is the genus of the curves in the system. This extends a previous result of Castelnuovo and Enriques. We classify linear systems whose dimension belongs to certain intervals which naturally arise from Castelnuovo’s theorem. Then we make an application to the following moduli problem: what is the maximum number of moduli of curves of geometric genus g varying in a linear system on a surface? It turns out that, for g22, the answer is 2g+1, and it is attained by trigonal canonical curves varying on a balanced rational normal scroll.

Article information

Source
Kyoto J. Math., Volume 51, Number 3 (2011), 633-645.

Dates
First available in Project Euclid: 1 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1312205241

Digital Object Identifier
doi:10.1215/21562261-1299909

Mathematical Reviews number (MathSciNet)
MR2824002

Zentralblatt MATH identifier
1226.14047

Subjects
Primary: 14C20: Divisors, linear systems, invertible sheaves
Secondary: 14J26: Rational and ruled surfaces

Citation

Castorena, Abel; Ciliberto, Ciro. On a theorem of Castelnuovo and applications to moduli. Kyoto J. Math. 51 (2011), no. 3, 633--645. doi:10.1215/21562261-1299909. https://projecteuclid.org/euclid.kjm/1312205241


Export citation

References

  • [1] R. D. M. Accola, On Castelnuovo’s inequality for algebraic curves, I, Trans. Amer. Math. Soc. 251 (1979), 357–373.
  • [2] M. Beltrametti and A. Sommese, “Zero cycles and kth order embeddings of smooth projective surfaces” with an appendix by Lothar Göttsche, in Problems in the Theory of Surfaces and Their Classification (Cortona, Italy, 1988), Sympos. Math. 32, Academic Press, London, 1991, 33–48.
  • [3] A. Calabri and C. Cliberto, Birational classification of curves on rational surfaces, to appear in Nagoya Math. J., preprint, arXiv:0906.4963v1 [math.AG]
  • [4] G. Castelnuovo, Massima dimensione dei sistemi lineari di curve piane di dato genere. Annali di Mat. (2) 18 (1890), 119–128.
  • [5] G. Castelnuovo, Ricerche generali sopra i sistemi lineari di curve piane. Mem. R. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (2) 42 (1890–1891), 137–188.
  • [6] C. Ciliberto, “Alcune applicazioni di un classico procedimento di Castelnuovo” in Seminari di Geometria, Università di Bologna, Istituto di Geometria, Dipartimento di Matematica, 1982–83, 17–43.
  • [7] C. Ciliberto, “Linear systems of curves on a surface and varieties of secant spaces” (in Italian) in Luigi Cremona (1830–1903), Incontr. Studio 36, Istituto Lombardo di Scienze e Lettere, Milano, 2005.
  • [8] C. Ciliberto and G. van der Geer, Subvarieties of the moduli space of curves parameterizing Jacobians with nontrivial endomorphisms, Amer. J. Math. 114 (1991), 551–570.
  • [9] C. Ciliberto and R. Lazarsfeld, “On the uniqueness of certain linear series on some classes of curves” in Complete Intersections (Acireale, Italy, 1983), Lecture Notes in Math. 1092, Springer, Berlin, 1984, 198–213.
  • [10] C. Ciliberto and F. Russo, Varieties with minimal secant degree and linear systems of maximal dimension on surfaces, Adv. Math. 200 (2006), 1–50.
  • [11] F. Enriques, Sulla massima dimensione dei sistemi lineari di dato genere appartenenti a una superficie algebrica, Atti Reale Acc. Scienze Torino 29 (1894), 275–296.
  • [12] P. Griffiths and J. Harris, Algebraic geometry and local differential geometry, Ann. Sci. École Norm. Sup. 12 (1979), 355–452.
  • [13] A. Maroni, Le serie lineari sulle curve trigonali, Ann. Mat. Pura Appl. (4) 25 (1946), 343–354.
  • [14] M. Reid, Surfaces of small degree, Math. Ann. 275 (1986), 71–80.
  • [15] I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), 309–316.