Plane curves as Pfaffians

TL;DR

This paper characterizes all linear Pfaffian representations of smooth plane curves by linking them to specific rank 2 vector bundles on the curve, providing an explicit parametrization within the moduli space.

Contribution

It establishes a correspondence between Pfaffian representations of plane curves and rank 2 vector bundles with canonical determinant, enriching the understanding of their geometric structure.

Findings

Parametrization of Pfaffian representations via moduli space

Explicit correspondence between Pfaffian representations and vector bundles

Identification of an open subset in the moduli space for these representations

Abstract

Let $C$ be a smooth curve in $\PP^2$ given by an equation F=0 of degree $d$. In this paper we parametrise all linear pfaffian representations of $F$ by an open subset in the moduli space $M_C(2,K_C)$. We construct an explicit correspondence between pfaffian representations of $C$ and rank 2 vector bundles on $C$ with canonical determinant and no sections.