Pfaffian representations of plane curves

TL;DR

This paper provides explicit linear Pfaffian representations for homogeneous polynomials defining smooth plane curves of degree up to 25 over any commutative ring, extending Beauville's result beyond algebraically closed fields.

Contribution

It introduces an empirical method to find Pfaffian representations for plane curves of degree up to 25, generalizing Beauville's existence result to broader algebraic settings.

Findings

Explicit Pfaffian representations for degrees up to 25

Empirical method for constructing Pfaffian representations

Extension of Beauville's result to arbitrary commutative rings

Abstract

Let R be a commutative ring with 1. For every homogeneous polynomial f(X_0,X_1,X_2) in R[X_0,X_1,X_2] of degree d <= 25, we find a explicit linear Pfaffian R-representation of f. We describe an empirical method that leads us to find such R-representations. This generalizes and constitutes an alternative proof (up to degree 25) of a result due to A. Beauville [Bea] about the existence of linear Pfaffian K-representations for any smooth plane curve of degree d >= 2, where K is an algebraically closed field of characteristic zero.