Complete description of determinantal representations of smooth irreducible curves

Abstract

Determinantal representations of algebraic curves are interesting in themselves, and their classification is equivalent to the simultaneous classification of triples of matrices. We present a complete description of determinantal representations of smooth irreducible curves over any algebraically closed field. We used the notion of the class of divisors of the vector bundle corresponding to a determinantal representation; we prove that two determinantal representations of a smooth curve F are equivalent if and only if the classes of divisors of the corresponding vector bundles coincide. We give a precise characterization of those classes of divisors that arise from vector bundles corresponding to determinantal representations of F. Then we obtain a parametrization of determinantal representations of F, up to equivalence, by the points of the Jacobian variety of F not on some exceptional subvariety. In particular it follows that any smooth curve of order 3 or greater possesses an infinite number of nonequivalent determinantal representations. We also specialize our results to symmetrical and self-adjoint representations.