Inversion of Abelian integrals on small genus curves
type: Item
instance of: scholarly article
title: Inversion of Abelian integrals on small genus curves
zbMATH Open document ID: 0579.14023
DOI: 10.1007/BF01458456
author: Kevin R. Coombes
author: Robert J. jun. Fisher
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164142
review text: Let M be a smooth projective curve of genus \(g>0\). Let J be the Jacobian of M and let \(M^{(i)}\) be the i-th symmetric product. Fix a base point \(p\in M\) and let \(\phi_ i: M^{(i)}\to J\) be the Abel-Jacobi map. For \(i\geq 2g-1\), Mattuck has shown that \(\phi_ i: M^{(i)}\to J\) is a \({\mathbb{P}}^{i-g}\)-bundle. The most interesting case occurs when \(i=2g-1\), since the bundles for larger i are determined by this one [see the book of \textit{R. C. Gunning} on this subject entitled ''Riemann surfaces and generalized theta functions'' (1976; Zbl 0341.14013)]. The inversion of abelian integrals problem asks: What is an explicit description of the transition functions of the bundle \(\phi_ i ?\) In this paper we present a solution to this question when M has genus 2 and when M is non-hyperelliptic of genus 3. For further background of this problem, see the cited book as well as the articles of Gunning, Kempf and Mattuck.
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 3927085
zbMATH Keywords: Jacobian
zbMATH Keywords: Abel-Jacobi map
zbMATH Keywords: inversion of abelian integrals problem
zbMATH Keywords: genus
Wikidata QID: Q57556841
MaRDI profile type: Publication
cites work: Q4108448
cites work: On Generalized Theta Functions
cites work: Toward the inversion of abelian integrals. I
cites work: Toward the Inversion of Abelian Integrals, II
cites work: Inversion of Abelian integrals
cites work: Picard bundles
cites work: Symmetric Products and Jacobians
cites work: Q5628266
description: scientific article; zbMATH DE number 3927085