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

TL;DR

This paper verifies the Batyrev-Manin and Batyrev-Tschinkel conjectures for a specific non-normal cubic surface and its symmetric square, providing point counts and geometric insights over any number field.

Contribution

It completes the point count analysis for a non-normal cubic surface and its symmetric square, confirming conjectural growth rates and constants in these cases.

Findings

Point count matches Batyrev-Manin conjecture growth rate.

Constant reflects the variety's geometry as predicted by Batyrev-Tschinkel.

Main term of the symmetric square's point count is explained, with some conditions.

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.