A short note on Cayley-Salmon equations

TL;DR

This paper explores Cayley-Salmon equations for smooth cubic surfaces, detailing their classical derivation, explicit calculations, applications to specific models, and connections to recent computational methods.

Contribution

It provides a classical proof, explicit calculations, and links to modern computational approaches for Cayley-Salmon equations on cubic surfaces.

Findings

120 distinct Cayley-Salmon equations for a smooth cubic surface

Explicit calculation methods demonstrated on Clebsch surface

Connection established with recent computational work by Cueto and Deopurkar

Abstract

A Cayley-Salmon equation for a smooth cubic surface $S$ in $\mathbb P^3$ is an expression of the form $l_1l_2l_3 - m_1m_2m_3 = 0$ such that the zero set is $S$ and $l_i$, $m_j$ are homogeneous linear forms. This expression was first used by Cayley and Salmon to study the incidence relations of the 27 lines on $S$. There are 120 essentially distinct Cayley-Salmon equations for $S$. In this note we give an exposition of a classical proof of this fact. We illustrate the explicit calculation to obtain these equations and we apply it to Clebsch surface and to the octanomial model. Finally we show that these $120$ Cayley-Salmon equations can be directly computed using recent work by Cueto and Deopurkar.