Rationality and parametrizations of algebraic curves under specializations
type: Item
instance of: scholarly article
title: Rationality and parametrizations of algebraic curves under specializations
author: Sebastian Falkensteiner
author: J. Rafael Sendra
published in: Journal of Algebra
publication date: 2024-09-30T00:00:00Z
review text: The authors study the properties of an algebraic curve that depends on parameters \(a_1,\dots,a_n\), i.e., the zero-set of a polynomial \(F(x,y)\) with \(F\in\mathbb{K}(a_1,\dots,a_n)[x,y]\), where \(\mathbb{K}\) denotes a computable field of characteristic~\(0\) and where \(F(x,y)\) is assumed to be irreducible over the algebraic closure of \(\mathbb{K}(a_1,\dots,a_n)\). Clearly, certain algebraic properties of the curve defined by~\(F\) that hold for generic values of the parameters, such as the genus of the curve or reducibility, can change when \(a_1,\dots,a_n\) are instantiated by special values. One goal of the paper is to identify (Zariski-closed) sets in the parameter space where non-generic behavior may appear. This leads to a partition of the parameter space, so that in each subset of the partition the specialized curve is either reducible or its (geometric) genus is invariant. In the case of genus-zero curves, i.e., curves that admit a rational parametrization, the parameter space can be further subdivided into regions where a given parametrization (for generic parameters) specializes properly. As applications of their results, the authors discuss (i) level curves of a surface (ii) linear homotopy deformation of two curves, and (iii) rational solutions of functional algebraic equations.
reviewed by: Christoph Koutschan
Mathematics Subject Classification ID: 14Q05
zbMATH DE Number: 7920100
zbMATH Keywords: algebraic curves
zbMATH Keywords: parameters
zbMATH Keywords: rational parametrizations
zbMATH Keywords: singularities
zbMATH Keywords: geometric genus
zbMATH Keywords: Hilbert's irreducibility theorem
MaRDI profile type: Publication
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DOI: 10.1016/J.JALGEBRA.2024.07.009
description: scientific article; zbMATH DE number 7920100