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Let $C$ be a smooth curve in $\PP^2$ given by an equation F=0 of degree $d$.In this paper we consider elementary transformations of linear pfaffianrepresentations of $C$. Elementary transformations can be interpreted asactions on a rank 2 vector bundle on $C$ with canonical determinant and nosections, which corresponds to the cokernel of a pfaffian representation. Everytwo pfaffian representations of $C$ can be bridged by a finite sequence ofelementary transformations. Pfaffian representations and elementarytransformations are constructed explicitly. For a smooth quartic, applicationsto Aronhold bundles and theta characteristics are given.

Elementary Transformations of Pfaffian Representations of Plane Curves