Advertisement

Geometriae Dedicata

, Volume 129, Issue 1, pp 89–99 | Cite as

A speciality theorem for curves in P5

  • Vincenzo Di Gennaro
  • Davide Franco
Original Paper

Abstract

Let \(C{\subset}\,{\bf P}^r\) be an integral projective curve. One defines the speciality index e(C) of C as the maximal integer t such that \(h^0(C,\omega_C(-t)) > 0\) , where ω C denotes the dualizing sheaf of \(C\) . Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if \(C {\subset}\,{\bf P}^5\) is an integral degree d curve not contained in any surface of degree  < s, in any threefold of degree  < t, and in any fourfold of degree  < u, and if \(d{ > > }s{ > > } t > > u\geq 1\) , then \( e(C)\leq {\frac{d}{s}}+{\frac{s}{t}}+{\frac{t}{u}}+u-6. \) Moreover equality holds if and only if C is a complete intersection of hypersurfaces of degrees u, \({\frac{t}{u}}\) , \({\frac{s}{t}}\) and \({\frac{d}{s}}\) . We give also some partial results in the general case \(C\subset {\bf P}^r\) , \(r\geq 3\) .

Keywords

Complex projective curve Speciality index Arithmetic genus Adjunction formula Complete intersection Linkage Castelnuovo - Halphen Theory Flag conditions 

Mathematics Subject Classification (2000)

Primary 14N15 14H99 14M10 Secondary 14M06 14N30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Chiantini L., Ciliberto C. and Di Gennaro V. (1993). The genus of projective curves. Duke Math. J. 70(2): 229–245 MATHCrossRefMathSciNetGoogle Scholar
  2. Chiantini L., Ciliberto C. and Di Gennaro V. (1995). The genus of curves in P 4 verifying certain flag conditions. Manuscripta Math. 88: 119–134 MATHCrossRefMathSciNetGoogle Scholar
  3. Chiantini L., Ciliberto C. and Di Gennaro V. (1996). On the genus of projective curves verifying certain flag conditions. Boll. U.M.I. (7)(10-B): 701–732 MathSciNetGoogle Scholar
  4. Ciliberto C. and Di Gennaro V. (2004). Factoriality of certain threefolds complete intersection in P 5 with ordinary double points. Commun. Algebra 32(7): 2705–2710 MATHCrossRefMathSciNetGoogle Scholar
  5. Di Gennaro, V.: Hierarchical structure of the family of curves with maximal genus verifying flag conditions. Proc. Amer. Math. Soc. (to appear)Google Scholar
  6. Eisenbud, D., Harris J.: Curves in projective space Sém. Math. Sup. 85, Les Presses du l’Université de Montréal, Montréal 1982Google Scholar
  7. Franco D., Kleiman S.L. and Lascu A.T. (2000). Gherardelli linkage and complete intersections. Michigan Math. J. 48: 271–279 MATHCrossRefMathSciNetGoogle Scholar
  8. Gruson, L., Peskine, C.: Genre des courbes dans l’espace projectif. Algebraic Geometry: Proceedings, Norway, 1977, Lecture Notes in Math., Springer-Verlag, New York 687 31–59 (1978)Google Scholar
  9. Migliore, J.C.: Introduction to Liaison Theory and Deficiency Modules. Birkhäuser, Prog. Math. 165 (1998)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly
  2. 2.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”Università di Napoli “Federico II”NapoliItaly

Personalised recommendations