Explicit constructions for genus 3 Jacobians
type: Item
instance of: scholarly article
title: Explicit constructions for genus 3 Jacobians
zbMATH Open document ID: 1306.14014
DOI: 10.1216/RMJ-2014-44-4-1367
author: Jesús Romero-Valencia
author: Alexis G. Zamora
published in: Rocky Mountain Journal of Mathematics
publication date: 2014-11-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0904.4537
full work available at URL: https://projecteuclid.org/euclid.rmjm/1414760958
review text: Let \(X\) be a nonsingular curve of genus three defined by a quartic \(F\) in \(\mathbb{P}^2\). Assume that \(X\) has a point \(p\) such that its tangent line intersects \(X\) at \(p\) with multiplicity four. Under this assumption the authors show in this paper that there exists a bijection between an open subset of the jacobian variety of \(X\) and the variety \(\{(A, B, G, H)\in H^0(\mathcal{O}_{\mathbb{P}^2}(2))^4;F=AG-BH\}\) with \(A\) and \(B\) of the form \(A=a_{00}+a_{10}x+a_{01}y-x^2, B=b_{00}+b_{10}x+b_{01}y-xy\), and \(H\) without the term \(x^2\). Furthermore they provide us with an explicit form of the morphism associated with the multiplication by -1, and a set of equations for the Kummer variety of the jacobian variety.
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14H50
zbMATH DE Number: 6368770
zbMATH Keywords: Jacobian varieties
zbMATH Keywords: curves of genus three
MaRDI profile type: Publication
arXiv ID: 0904.4537
cites work: Determinantal hypersurfaces.
cites work: The Geometry of Syzygies
cites work: Q4697470
OpenAlex ID: W2074532816
description: scientific article; zbMATH DE number 6368770