The discriminant of a cubic surface

TL;DR

This paper constructs explicit cubic surfaces over rationals with Galois group actions on lines corresponding to the simple group of order 25920, analyzing their discriminants and rational points.

Contribution

It provides explicit examples of cubic surfaces with maximal Galois group action and studies the discriminant's relation to this group, including rational point analysis.

Findings

Explicit examples of cubic surfaces with Galois group of order 25920

Relation between discriminant and Galois group action

Construction of an accumulating subvariety in the parameter space

Abstract

We construct explicit examples of cubic surfaces over $\bbQ$ such that the 27 lines are acted upon by the index two subgroup of the maximal possible Galois group. This is the simple group of order $25 920$. Our examples are given in pentahedral normal form with rational coefficients. For such cubic surfaces, we study the discriminant and show its relation to the index two subgroup. On the corresponding parameter space, we search for rational points, discuss their asymptotic, and construct an accumulating subvariety.