Elementary transformations of determinantal representations of algebraic curves

Abstract

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.