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arXiv · 2011.14466

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

Nils Gubela·Julian Lyczak
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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.

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Nils Gubela, Julian Lyczak. 2020-11-29. The Batyrev-Tschinkel conjecture for a non-normal cubic surface and its symmetric square. https://arxiv.org/abs/2011.14466

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