Elementary transformations of determinantal representations of algebraic curves
type: Item
instance of: scholarly article
title: Elementary transformations of determinantal representations of algebraic curves
zbMATH Open document ID: 0732.14011
DOI: 10.1016/0024-3795(90)90113-Q
author: Victor Vinnikov
published in: Linear Algebra and its Applications
publication date: 1990-01-01T00:00:00Z
review text: A determinantal representation of a projective plane curve F(x) of degree \(n\) is a matrix U(x) of order \(n\) whose entries are linear forms such that \(\det (U(x))=cF(x)\) for some constant c. Two determinantal representations \(U_ 1, U_ 2\) are called equivalent if there exist constant matrices M, N such that \(U_ 2=MU_ 1N\). If F(x) is a smooth curve, ker(U(x)) is a vector bundle of rank 1. Up to equivalence, the determinantal representations of F(x) are parametrized, via the class of divisors of the corresponding vector bundle, by the points of the Jacobian variety of F(x) not on the exceptional subvariety [see \textit{R. J. Cook} and \textit{A. D. Thomas}, Q. J. Math., Oxf. II. Ser. 30, 423-429 (1979; Zbl 0437.14004)]. Based on this fact, the author shows that one can build up all the non-equivalent determinantal representations of F(x) by applying a finite sequence of the so-called elementary transformations to F(x) whose number is bounded by the genus of F(x).
reviewed by: Trung Ngo Viet
Mathematics Subject Classification ID: 14H99
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 4210316
zbMATH Keywords: determinantal representation of a projective plane curve
zbMATH Keywords: divisors
zbMATH Keywords: Jacobian variety
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4210316