Explicit linear pfaffian representations of plane curves up to degree 5

TL;DR

This paper demonstrates that all homogeneous polynomials of degree up to 5 in three variables over a commutative ring have explicit linear Pfaffian representations, extending Beauville's results to degrees up to 5.

Contribution

It provides explicit, concise Pfaffian representations for plane curves of degree up to 5, generalizing previous results for smooth curves.

Findings

Explicit Pfaffian representations for degrees 3, 4, 5

Short self-contained proof of existence

Extension of Beauville's result to degree 5

Abstract

Let R be a commutative ring with 1. We prove that every homogeneous polynomial f(x_0,x_1,x_2) in R[x_0,x_1,x_2] up to degree 5 admits a linear Pfaffian R-representation. We believe that conceptually we give the shortest self-contained proof possible: we exhibit explicitly such a representation. In this sense, we generalize (up to degree 5) a result due to A. Beauville about the existence of Pfaffian representations for any smooth plane curve of any degree.