  For each integer $D\ge3$, we give a sharp bound on the number of lines
contained in a smooth complex $2D$-polarized $K3$-surface in
$\mathbb{P}^{D+1}$. In the two most interesting cases of sextics in
$\mathbb{P}^4$ and octics in $\mathbb{P}^5$, the bounds are $42$ and $36$,
respectively, as conjectured in an earlier paper.
