A characterization of hyperelliptic Jacobians
type: Item
instance of: scholarly article
title: A characterization of hyperelliptic Jacobians
zbMATH Open document ID: 0798.14016
DOI: 10.1007/BF02568350
author: Giambattista Marini
published in: Manuscripta Mathematica
publication date: 1994-11-03T00:00:00Z
full work available at URL: https://eudml.org/doc/155846
review text: Let \(C\) be a smooth curve of genus \(g\), \(J(C)\) its Jacobian and \(\Theta\) the Theta divisor on \(J(C)\). It is a well known result that the dimension of the singularity locus \(\text{sing} (\Theta)\) is \(g-4\) for nonhyperelliptic \(C\) and \(g-3\) for hyperelliptic curves \(C\). This statement is part of a criterion characterizing Jacobians of hyperelliptic curves among all principally polarized abelian varieties. The main theorem of the paper under consideration is: A principally polarized abelian variety \((X,\Theta)\) of dimension \(g\) is the Jacobian of a hyperelliptic curve if and only if \(\dim \text{sing} (\Theta) \leq g-3\) and there exists a global section of the sheaf \({\mathcal O}_ \Theta (\Theta)\) whose zero locus admits nonreduced components.
Mathematics Subject Classification ID: 14H42
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14K25
zbMATH DE Number: 492752
zbMATH Keywords: Theta divisor on Jacobian
zbMATH Keywords: dimension of the singularity locus
zbMATH Keywords: characterizing Jacobians of hyperelliptic curves among all principally polarized abelian varieties
zbMATH Keywords: principally polarized abelian variety
reviewed by: Christina Birkenhake
MaRDI profile type: Publication
cites work: Q3788123
cites work: Another proof of a conjecture of S. P. Novikov on periods of Abelian integrals on Riemann surfaces
cites work: Q5287185
cites work: Theta functions on Riemann surfaces
cites work: Characterization of Jacobian varieties in terms of soliton equations
cites work: A criterion for Jacobi varieties
description: scientific article; zbMATH DE number 492752