7. Projective models in smooth scrolls

Abstract

Let D be a free Clifford divisor on a non-Clifford general polarized K3 surface S. Assume that $\mathcal{T} = \mathcal{T}(c,D) = \mathcal{T}(c,D,\{D_{\lambda} \})$ is smooth. This is equivalent to the conditions D ² = 0 and $\mathcal{R}_{L,D} = \emptyset$ when D is perfect. In any case these two conditions are necessary to have $\mathcal{T}$ smooth, so |D| has projective dimension 1 and the pencil $D_{\lambda}$ is uniquely determined. We recall that $\varphi_L(S)$ is denoted by S’.