The number of cubic surfaces with 27 lines over a finite field

Abstract

We determine the number of cubic surfaces with 27 lines over a finite field F-q. This is based on exploiting the relationship between non-conical six-arcs in a projective plane embedded in projective three-space and cubic surfaces with 27 lines. We revisit this classical relationship, which goes back to work of Clebsch in the nineteenth century. Our result can be used as an enumerative check for a computer classification of cubic surfaces with 27 lines over finite fields.