A speciality theorem for curves in P5 contained in Noether–Lefschetz general fourfolds

Let C⊂ Pr be an integral projective curve. We define the speciality index e(C) of C as the maximal integer t such that h0(C, ωC(- t)) > 0 , where ωC denotes the dualizing sheaf of C. In the present paper we consider C⊂ P5 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)≤dsnk+s+n+k-6. Moreover equality holds if and only if C is a complete intersection of T: = F∩ G with hypersurfaces of degrees s and dsnk.