Projective normality of abelian surfaces given by primitive line bundles
type: Item
instance of: scholarly article
title: Projective normality of abelian surfaces given by primitive line bundles
zbMATH Open document ID: 0943.14022
published in: Manuscripta Mathematica
publication date: 2000-09-03T00:00:00Z
review text: Let \(A\) be an abelian variety with \(\dim A=g\) and let \(L\) be an ample line bundle of type \(\delta=(\delta_{1},\dots , \delta_{g})\) on \(A\). Then the embedding of \(A\) given by the morphism defined by \(L^{n}\) is projectively normal if \(n\geq 3\) or \(n=2\) and no point of \(K(L^{2})\) is a base point for \(L\), where \(K(L^{2})=\{a\in A \mid L\cong t_{a}^{*}L\}\) and \(t_{a}\) is the translation map by \(a\) on \(A\). In this paper, the author studies the case of an abelian surface when \(L\) is primitive, that is, \(L\) is of the type \((1, \delta)\). If \(\delta\leq 4\), then \(L\) is not very ample. If \(\delta=5\) or \(6\), then the embedding defined by the linear system \(|L|\) is not projectively normal. So the author considers the case where \(\delta\geq 7\) and obtains the following theorem: Let \(L\) be an ample line bundle of type \((1, \delta)\) on an abelian surface \(A\). Suppose the Néron-Severi group of \(A\) is isomorphic to \(\mathbb{Z}\) and \(\delta\geq 7\). Then the image of \(A\), under the morphism defined by the linear system \(|L|\), is projectively normal. This case is the generic situation of \textit{G. P. Pirola} [Math. Ann. 282, No. 3, 361-368 (1988; Zbl 0625.14024)]. This result can be proved by an analysis of the representations of the theta group \(\mathcal{G}(L)\) on \(\text{Sym}^{2}H^{0}(L)\) as well as on \(H^{0}(L^{2})\).
reviewed by: Yoshiaki Fukuma
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 1275769
zbMATH Keywords: abelian surface
zbMATH Keywords: projective normality
zbMATH Keywords: primitive line bundle
zbMATH Keywords: theta group
author: Jaya N. Iyer
MaRDI profile type: Publication
full work available at URL: s002290050131
OpenAlex ID: W2374863308
DOI: 10.1007/S002290050131
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description: scientific article; zbMATH DE number 1275769