A speciality theorem for curves in P5

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