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

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 N'eron-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 $e©\leq {\frac{d}{snk}}+s+n+k-6.$Moreoverequalityholdsifandonlyif$C$ is a complete intersection of $T:=F\cap G$ with hypersurfaces of degrees $s$ and $\frac{d}{snk}$.