Effective criteria for bigraded birational maps. (July 2017)

• Record Type: Journal Article
• Authors: Botbol, Nicolás; Busé, Laurent; Chardin, Marc; Hassanzadeh, Seyed Hamid; Simis, Aron; Tran, Quang Hoa
• Is Part Of: Journal of symbolic computation. Volume 81(2017)
• Page Start: 69
• Page End: 87
• Publication Date: 2017-07
• Subjects: Bigraded rational maps -- Jacobian dual rank -- Birationality criteria -- Bigraded base ideal -- Rees algebras; 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.2016.12.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this paper, we consider rational maps whose source is a product of two subvarieties, each one being embedded in a projective space. Our main objective is to investigate birationality criteria for such maps. First, a general criterion is given in terms of the rank of a couple of matrices that became to be known as Jacobian dual matrices . Then, we focus on rational maps from P 1 × P 1 to P 2 in very low bidegrees and provide new matrix-based birationality criteria by analyzing the syzygies of the defining equations of the map, in particular by looking at the dimension of certain bigraded parts of the syzygy module. Finally, applications of our results to the context of geometric modeling are discussed at the end of the paper.