A speciality theorem for curves in ${\mathbb {P}}^5$ contained in Noether–Lefschetz general fourfolds

Abstract

Let $C\subset {\mathbb {P}}^r$ be an integral projective curve. We define 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. In the present paper we consider $C\subset {\mathbb {P}}^5$ an integral degree d curve and we denote by s the minimal degree for which there exists a hypersurface of degree s containing C. We assume that C is contained in two smooth hypersurfaces F and G, with $deg(F)=n>k=deg (G)$. We assume additionally that F is Noether–Lefschetz general, i.e. that the 2-th Néron–Severi group of F is generated by the linear section class. Our main result is that in this case the speciality index is bounded as $e(C)\le {\frac{d}{snk}}+s+n+k-6.$ Moreover equality holds if and only if C is a complete intersection of $T:=F\cap G$ with hypersurfaces of degrees s and ${\frac{d}{snk}}$.