Complete description of determinantal representations of smooth irreducible curves Academic Article uri icon

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.

publication date

  • January 1, 1989