Pfaffian representations of cubic surfaces

TL;DR

This paper presents an algorithm for constructing linear Pfaffian representations of cubic surfaces over fields of characteristic zero, providing explicit methods and examples for such representations.

Contribution

It introduces a new algorithm for obtaining Pfaffian representations of cubic surfaces and proves their existence over certain algebraic extensions.

Findings

Algorithm for Pfaffian representation given a cubic surface and a point on it.

Existence of Pfaffian representations over algebraic extensions of degree at most six.

Explicit example demonstrating the construction.

Abstract

Let K be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial F of degree three in K[x_0,x_1,x_2,x_3] and a zero A of F in P^3_K and ensures a linear pfaffian representation of V(F) with entries in K[x_0,x_1,x_2,x_3], under mild assumptions on F and A. We use this result to give an explicit construction of (and to prove the existence of) a linear pfaffian representation of V(F), with entries in K'[x_0,x_1,x_2,x_3], being K' an algebraic extension of K of degree at most six. An explicit example of such a construction is given.