Primitive line bundles on abelian threefolds
type: Item
instance of: scholarly article
title: Primitive line bundles on abelian threefolds
zbMATH Open document ID: 0807.14030
DOI: 10.1007/BF02567860
author: Herbert Lange
author: S. Ramanan
author: Christina Birkenhake
published in: Manuscripta Mathematica
publication date: 1995-03-05T00:00:00Z
full work available at URL: https://eudml.org/doc/155902
review text: Let \(L\) be an ample line bundle of polarization type \((1, d_ 2, \dots, d_ n)\) on the abelian \(n\)-fold \(X\). It is known that if \(n = 2\) then \(L\) is very ample iff \(d = d_ 2 \geq 5\), and \(L\) is globally generated (free) iff \(d \geq 3\). If \(n \geq 3\) not much is known. In this paper the authors treat the case \(n = 3\). The general Ein-Lazarsfeld criterion for freeness of an ample line bundle \(L\) on a 3-fold \(X\) assumes the non-existence of some special curves on \(X\). The question is to replace this criterion by an effective one in the case of abelian 3-folds. Example 4.2 shows that no numerical criterion guarantees the freeness of \(L\): see proposition 4.3. However, the pair \((X,L)\) defines at least one principally polarized abelian 3-fold \(Y\) isogenous to \(X\). In this paper, the authors find such sufficient conditions for freeness of \(L\) only in terms of the fixed isogeny \(X \to Y\), and depending of the decomposition type of \(Y\) (being a product of 1, 2, or 3 irreducible abelian varieties): see theorem 1. -- Next, the authors study sufficient conditions ensuring very ampleness of \(L\), by using the method of Comessatti for Jacobians with two principal polarizations, and Sakai's version of Reider's theorem yielding very ampleness for a large class of line bundles \(L\): see theorem 2. In particular, if the polarization is of type \((1,1,d)\), \(d \geq 13\), \(d \neq14\), then \(L\) is very ample.
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14K99
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J40
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 554386
zbMATH Keywords: abelian variety
zbMATH Keywords: freeness of an ample line bundle on a 3-fold
zbMATH Keywords: ample line bundle
zbMATH Keywords: isogeny
zbMATH Keywords: very ampleness
reviewed by: Atanas Iliev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 554386