Higher order embeddings of algebraic surfaces of Kodaira dimension zero (Q1297952)

Let \(L\) be a line bundle on a smooth connected projective surface \(S\) over the complex numbers. \(L\) is said to be \(k\)-very ample (resp. \(k\)-spanned) for an integer \(k\geq 0\) if, given any (resp. curvilinear) 0-cycle \((Z,{\mathcal O}_Z)\) of \(S\) with \(\text{length} ({\mathcal O}_Z)=k+1\), the restriction map \(\Gamma(L)\to \Gamma ({\mathcal O}_Z(L))\) is surjective. In this paper, we study the \(k\)-very ampleness and \(k\)-spannedness of line bundles on a minimal surface of Kodaira dimension zero. Given a line bundle \(L\), the question of when \(L\) is \(k\)-very ample (resp. \(k\)-spanned) breaks up naturally into the cases when \(L\cdot L\geq 4k+6\) and the case when \(L\cdot L\leq 4k+4\). If \(L\cdot L\geq 4k+6\), then we give a numerical criterion of Reider-type for a nef line bundle \(L\) to be \(k\)-very ample (resp. \(k\)-spanned). On the other hand, the condition \(L\cdot L\leq 4k+4\) implies that the polarized surface \((S,L)\) with \(L\) \(k\)-very ample is either a minimal K3 surface with \(L\cdot L=4k\), \(4k+2\), \(4k+4\) or a minimal Enriques surface with \(L\cdot L=4k+4\). In the K3 surface case, we characterize a smooth curve \(C\in|L|\) by using its Clifford index and gonality.