Singularities of plane rational curves via projections. (May 2018)

• Record Type: Journal Article
• Authors: Bernardi, Alessandra; Gimigliano, Alessandro; Idà, Monica
• Is Part Of: Journal of symbolic computation. Volume 86(2018)
• Page Start: 189
• Page End: 214
• Publication Date: 2018-05
• Subjects: Rational curves -- Parameterizations -- Singularities -- Algorithms -- Projections; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2017.05.003
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We consider the parameterization f = ( f 0 : f 1 : f 2 ) of a plane rational curve C of degree n, and we study the singularities of C via such parameterization. We use the projection from the rational normal curve C n ⊂ P n to C and its interplay with the secant varieties to C n . In particular, we define viaf certain 0-dimensional schemes X k ⊂ P k, 2 ≤ k ≤ ( n − 1 ), which encode all information on the singularities of multiplicity ≥ k of C (e.g. using X 2 we can give a criterion to determine whether C is a cuspidal curve or has only ordinary singularities). We give a series of algorithms which allow one to obtain information about the singularities from such schemes.