A short note on Cayley-Salmon equations

Abstract

A Cayley-Salmon equation for a smooth cubic surface S in P³ is an expression of the form l₁l₂l₃ - m₁m₂m₃ = 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 the Clebsch surface and to the octanomial form appearing in work of Panizzut, Sertöz and Sturmfels. Finally we show that these 120 Cayley-Salom equations can be directly computed using recent work by Cueto and Deopurkar.