Lines on smooth polarized K3-surfaces

Abstract

For each integer D⩾3D⩾3, we give a sharp bound on the number of lines contained in a smooth complex 2D-polarized K3-surface in PD+1PD+1. In the two most interesting cases of sextics in P4P4 and octics in P5P5, the bounds are 42 and 36, respectively, as conjectured in an earlier paper.