A speciality theorem for curves in ${\mathbf{P}}^5$

Abstract

Let $C\subset {\mathbf{P}}^r$ 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 ${\omega }_C$ 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 $C\subset {\mathbf{P}}^5$ 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 $d>>s>>t>>u\ge 1$, then $e(C)\le \frac{d}{s}+\frac{s}{t}+\frac{t}{u}+u-6.$ 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 $C\subset {\mathbf{P}}^r$, $r\ge 3$.

10 pages