Elementary Transformations of Pfaffian Representations of Plane Curves

TL;DR

This paper studies elementary transformations of linear pfaffian representations of smooth plane curves, providing explicit constructions and applications to bundles and theta characteristics, showing all representations are interconnected through these transformations.

Contribution

It introduces explicit methods for elementary transformations of pfaffian representations and demonstrates their ability to connect any two such representations on a smooth curve.

Findings

Any two pfaffian representations of a smooth curve can be connected by elementary transformations.

Explicit constructions of pfaffian representations and transformations are provided.

Applications to Aronhold bundles and theta characteristics for quartic curves are discussed.

Abstract

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 pfaffian representations of $C$. Elementary transformations can be interpreted as actions on a rank 2 vector bundle on $C$ with canonical determinant and no sections, which corresponds to the cokernel of a pfaffian representation. Every two pfaffian representations of $C$ can be bridged by a finite sequence of elementary transformations. Pfaffian representations and elementary transformations are constructed explicitly. For a smooth quartic, applications to Aronhold bundles and theta characteristics are given.