The space of cubic surfaces equipped with a line

TL;DR

This paper computes the rational cohomology of the space of lines on smooth cubic surfaces over complex numbers, showing it matches that of PGL(4,C), and derives statistical properties of lines over finite fields.

Contribution

It explicitly calculates the cohomology of the space of lines on cubic surfaces and relates it to the cohomology of PGL(4,C), providing new insights into the topology and arithmetic of cubic surfaces.

Findings

Cohomology ring is isomorphic to that of PGL(4,C)

Average number of lines over finite fields approaches 1

Cohomology isomorphism holds in mixed Hodge structures

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.