A degree bound for the parametrization of a rational surface
type: Item
instance of: scholarly article
title: A degree bound for the parametrization of a rational surface
zbMATH Open document ID: 0961.14024
DOI: 10.1016/S0022-4049(98)00078-4
author: Josef Schicho
published in: Journal of Pure and Applied Algebra
publication date: 2001-05-28T00:00:00Z
review text: Parameterizations of rational surfaces are studied here in order to find an upper bound for the smallest degree of such a parameterization in terms of the degree or the sectional genus of the surface. The main results are the following: Let \(F\) be a rational surface of degree \(d\); then for any parameterization \(P\) of \(F\) we have \[ \deg P \geq \sqrt{{d\over 2}}. \] There exist a parameterization \(P\) of \(F\) such that \(\deg P \leq 2(d^2-4d+1)^2+2d+2\). If the sectional genus of \(F\) is \(p_1=0\), then there exists a parameterization \(P\) of \(F\) with \(\deg P\leq d\), while when \(p_1>0\) there exists \(P\) with \(\deg P \leq 4{2p_1-\deg F-1\choose 2}+4p_1\). Moreover, we always have \((\deg P-1)(\deg P-2)\geq 2p_1\), and, if \(p_1\geq 2\), \(d\leq 4p_1+4\). The main tool in proving the above result is repeated adjunction on the surface.
reviewed by: Alessandro Gimigliano
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14Q10
Mathematics Subject Classification ID: 14M20
zbMATH DE Number: 1398624
zbMATH Keywords: rational surfaces
zbMATH Keywords: sectional genus
zbMATH Keywords: degree of parameterization
zbMATH Keywords: adjunction
MaRDI profile type: Publication
Wikidata QID: Q126382658
full work available at URL: s0022-4049(98)00078-4
OpenAlex ID: W2154942664
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description: scientific article; zbMATH DE number 1398624