On sectional genus of \(k\)-very ample line bundles on smooth surfaces with non-negative Kodaira dimension
type: Item
instance of: scholarly article
title: On sectional genus of \(k\)-very ample line bundles on smooth surfaces with non-negative Kodaira dimension
zbMATH Open document ID: 0930.14003
author: Yoshiaki Fukuma
published in: Kodai Mathematical Journal
publication date: 1999-03-28T00:00:00Z
review text: Let \(C\) be a smooth complex projective surface polarized by an ample line bundle \(L\). In the paper under review the relations between the sectional genus \(g(L)\) and the irregularity \(q(X)\) are investigated under the assumption that \(L\) is \(k\)-very ample, i.e., for any 0-dimensional subscheme \((Z,{\mathcal O}_Z)\) with length \(({\mathcal O}_Z)\leq k+1\), the restriction homomorphism \(\Gamma(L)\to\Gamma(L\otimes{\mathcal O}_Z)\) is surjective. The author conjectures that if \(X\) has Kodaira dimension \(\kappa(X)\geq 0\) then \(g(L)\geq(k+2)q(X)\). To include the classical conjecture for polarized surfaces one needs to put \(k=-1\) if \(L\) is ample but not spanned by global sections. The conjectured inequality is proved for \(k\geq 0\) in the following cases: (i) \(\kappa(X)=0\), (ii) \(\kappa(X)=1\), and (iii) \(\kappa(X)=2\) and \(X\) fibres over a smooth curve \(C\) with \(q(X)\leq g(C)+1\). Moreover in cases (i) and (ii) the pairs \((X,L)\) for which \(g(L)=(k+2)q(X)\) are characterized.
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 1269765
zbMATH Keywords: \(k\)-very ample line bundle
zbMATH Keywords: sectional genus
zbMATH Keywords: irregularity
zbMATH Keywords: Kodaira dimension
zbMATH Keywords: polarized surfaces
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043870
OpenAlex ID: W2055126114
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DOI: 10.2996/KMJ/1138043870
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description: scientific article; zbMATH DE number 1269765