Higher order embeddings of algebraic surfaces of Kodaira dimension zero
type: Item
instance of: scholarly article
title: Higher order embeddings of algebraic surfaces of Kodaira dimension zero
zbMATH Open document ID: 0935.14005
DOI: 10.1007/PL00004664
author: Hiroyuki Terakawa
published in: Mathematische Zeitschrift
publication date: 2000-05-10T00:00:00Z
review text: 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.
reviewed by: Hiroyuki Terakawa
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14E25
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 1336832
zbMATH Keywords: spanned line bundle
zbMATH Keywords: very ample line bundle
zbMATH Keywords: minimal surface
zbMATH Keywords: Kodaira dimension
zbMATH Keywords: minimal K3 surface
zbMATH Keywords: minimal Enriques surface
zbMATH Keywords: Clifford index
zbMATH Keywords: gonality
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1336832