Multiplicity \(g\) points on theta divisors
type: Item
instance of: scholarly article
title: Multiplicity \(g\) points on theta divisors
zbMATH Open document ID: 0903.14017
DOI: 10.1215/S0012-7094-96-08214-9
author: Robert Varley
author: Roy Campbell Smith
published in: Duke Mathematical Journal
publication date: 1996-09-08T00:00:00Z
review text: Let \((A,\Theta)\) be a principally polarized abelian variety of dimension \(g\). \(\text{Sing}_m (\Theta)\) denotes the subvariety of the theta divisor \(\Theta\) of points of multiplicity \(\geq m\). -- \textit{J. Kollár} [``Shafarevich maps and automorphic forms'' (Princeton 1995; Zbl 0871.14015)] showed that the dimension of \(\text{Sing}_m (\Theta)\) is less or equal to \(g-m\) for \(m\geq 2\). This inequality is sharp as the case of products of elliptic curves shows. In remark 17.13.1 of the cited paper J. Kollár suggests that equality for some \(m\) implies a splitting of \((A,\Theta)\). The paper under review considers the case \(m=g\). It is shown that \(\text{Sing}_g (\Theta)\neq 0\) implies that \((A,\Theta)\) is a polarized product of \(g\) elliptic curves.
Mathematics Subject Classification ID: 14K25
zbMATH DE Number: 908692
zbMATH Keywords: singularities of the theta divisor
zbMATH Keywords: polarized product of elliptic curves
zbMATH Keywords: principally polarized abelian variety
reviewed by: Christina Birkenhake
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 908692