The space of cubic surfaces equipped with a line

Abstract

The Cayley--Salmon theorem implies the existence of a 27-sheeted covering space specifying lines contained in smooth cubic surfaces over $\mathbb{C}$. In this paper we compute the rational cohomology of the total space of this cover, using the spectral sequence in the method of simplicial resolution developed by Vassiliev. The covering map is an isomorphism in cohomology (in fact of mixed Hodge structures) and the cohomology ring is isomorphic to that of $PGL(4,\mathbb{C})$. We derive as a consequence of our theorem that over the finite field $\mathbb{F}_q$ the average number of lines on a cubic surface equals 1 (away from finitely many characteristics); this average is $1 + O(q^{-1/2})$ by a standard application of the Weil conjectures.