2016/11/17 by Di Gennaro, Vincenzo, Franco, Davide
#14H99 #14M06 #14M10 #14N15 #14N30 (Primary) #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1611.05631
Let C⊂ \mathbb 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⊂ \mathbb 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)≤ (d)/(snk)+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 (d)/(snk).