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

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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 Neron–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 \({\frac{d}{snk}}\).