Curves of the projective 3-space, tangent developables and partial spreads
type: Item
instance of: scholarly article
title: Curves of the projective 3-space, tangent developables and partial spreads
zbMATH Open document ID: 0992.51013
author: E. Ballico
author: Antonio Cossidente
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2002-09-12T00:00:00Z
review text: The authors are interested here in curves \(X\) of \(P(3,K)\) where \(K\) is an algebraic closed field of characteristic \(p \geq 0\) satisfying the following condition: (*) Tangent lines to \(X\) at distinct points are skew. In the case when \(K\) is the algebraic closure of \(GF(q)\), the above condition means that the tangent lines to \(X\) at \(GF(q)\)-rational points will form a (partial) spread of \(PG(3,q)\). Under suitable assumptions the authors prove that if the curve \(X\) satisfies the condition (*) then \(X\) must necessarily be a twisted cubic and they also find an infinite family of curves of \(PG(3,K)\) distinct from the twisted cubic, satisfying property (*). In order to do that, they characterize twisted cubic. More precisely, for a smooth degree \(d\) curve \(X\) such that for a general point \(P \in X\) there is no tangent line to \(X\) at a point \(Q\neq P\), with \(T_PX \cap T_QX \neq \empty\), they prove that \(X\equiv PG(1,K)\), and either \(d=3\) and \(X\) is a twisted cubic, or \(d=p^e+1\) and \(x\) is projectively equivalent to the rational curve \(D\) with the parametrization \[ (w_0,w_1)\mapsto (w_0^{p^e+1},w_0^{p^e}w_1,w_0w_1^{p^e},w_1^{p^e+1}). \] The authors also prove that two tangent lines to \(X\), where \(X\) is one of the above curves, are skew (if \(X\) is the curve \(D\), they also require \(p^e\geq 4\)).
reviewed by: Miriam Abdon
Mathematics Subject Classification ID: 51N35
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 1547318
zbMATH Keywords: twisted cubic
zbMATH Keywords: tangent developable
zbMATH Keywords: strange curve
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1547318