Elementary transformations of determinantal representations of algebraic curves
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).
determinantal representation of a projective plane curve
scientific article; zbMATH DE number 4210316