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