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


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 \(C\subset {\bf P}^r\) , \(r\geq 3\) .


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 


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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

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