On the preservation of \(k\)-very ampleness under adjunction
Let \(S\) be a smooth projective surface polarized by a very ample line bundle \(L\). The main lines of the classical adjunction theory run as follows: apart from a few well-understood exceptions \(K_ S \otimes L\) is spanned and big but as soon as \(S\) contains a \((-1)\)-line it fails to be very ample. Thus it is natural to look for another surface \(S'\) birational to \(S\) without \((-1)\)-lines. By contracting down all the \((- 1)\)-lines via a morphism \(r:S \to S'\) connected with the morphism defined by \(K_ S \otimes L\), we can replace the pair \((S,L)\) by its first reduction \((S',L')\), where \(L = r^*L'\). The basic result in adjunction theory for surfaces, due to Sommese and Van de Ven and independently to Serrano, claims that \(K_{S'} \otimes L'\) is very ample apart from four examples. On the other hand the authors have generalized the notion of very ampleness by introducing the \(k\)-very ample line bundles. Rather than a generalization, it is a kind of stratification since 1-very ampleness means very ampleness and \((k + 1)\)-very ampleness implies \(k\)-very ampleness. It is thus natural to see what happens in adjunction theory when the polarizing line bundle \(L\) is \(k\)-very ample. This is what is done in this article, namely the authors look for the \(k\)-very ampleness of \(K_ S \otimes L\) at least on a convenient reduction of the original pair \((S,L)\). The main tool used to this end is a Reider-type result, proved in a previous paper by the same authors. As happens in classical adjunction, the problem divides into two parts: the general case when the Reider-type result works and the case when \(L^ 2\) is small with respect to \(k\). The first step in this direction (case \(k = 2)\) had been done by Andreatta and the reviewer.
scientific article; zbMATH DE number 549732