Complete description of determinantal representations of smooth irreducible curves

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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.

Original languageEnglish
Pages (from-to)103-140
Number of pages38
JournalLinear Algebra and Its Applications
Volume125
Issue numberC
DOIs
StatePublished - 1 Jan 1989

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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