On the preservation of \(k\)-very ampleness under adjunction
type: Item
instance of: scholarly article
title: On the preservation of \(k\)-very ampleness under adjunction
zbMATH Open document ID: 0806.14015
DOI: 10.1007/BF02571657
author: Mauro C. Beltrametti
author: Andrew J. Sommese
published in: Mathematische Zeitschrift
publication date: 1994-10-10T00:00:00Z
full work available at URL: https://eudml.org/doc/174486
review text: 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.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 549732
zbMATH Keywords: adjoint line bundle
zbMATH Keywords: polarized smooth projective surface
zbMATH Keywords: \(k\)-very ample line bundles
zbMATH Keywords: adjunction theory
reviewed by: Marino Palleschi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 549732