The Batyrev-Tschinkel conjecture for a non-normal cubic surface and its symmetric square

Abstract

We complete the study of points of bounded height on irreducible non-normal cubic surfaces by doing the point count on the cubic surface $W$ given by $t_0^2{t}_2=t_1^2{t}_3$ over any number field. We show that the order of growth agrees with a conjecture by Batyrev and Manin and that the constant reflects the geometry of the variety as predicted by a conjecture of Batyrev and Tschinkel. We then provide the point count for its symmetric square ${\mathrm{Sym}}^{2}W$. Although we can explain the main term of the counting function, the Batyrev--Manin conjecture is only satisfied after removing a thin set. Finally we interpret the main term of the count on ${\mathrm{Sym}}^2({\mathbb{P}}^2\times {\mathbb{P}}^1)$ done by Le Rudulier using these conjecture.

19 pages