Elementary transformations of determinantal representations of algebraic curves

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We study elementary transformations, first introduced by Livsic and Kravitsky in an operator-theoretic context, of determinantal representations of algebraic curves. We consider determinantal representations of a smooth irreducible curve F over an algebraically closed field. When regarded as acting on the corresponding vector bundle the elementary transformations are a matrix generalization of scalar linear fractional transformations: they add to the class of divisors of the vector bundle a single zero and a single pole. We show that given a determinantal representation of F, we can build all the nonequivalent determinantal representations of F by applying to the given representation finite sequences of at most g elementary transformations, where g is the genus of F.

Original languageEnglish
Pages (from-to)1-18
Number of pages18
JournalLinear Algebra and Its Applications
Issue numberC
StatePublished - 1 Jan 1990

ASJC Scopus subject areas

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


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