Eventual tightness of projective dimension growth bounds: quadratic in the degree

Abstract

In projective dimension growth results, one bounds the number of rational points of height at most $H$ on an irreducible hypersurface in ${\mathbb{P}}^n$ of degree $d>3$ by $C(n)d^2{H}^{n-1}(\log H{)}^{M(n)}$, where the quadratic dependence in $d$ has been recently obtained by Binyamini, Cluckers and Kato in 2024 [1]. For these bounds, it was already shown by Castryck, Cluckers, Dittmann and Nguyen in 2020 [3] that one cannot do better than a linear dependence in $d$. In this paper we show that, for the mentioned projective dimension growth bounds, the quadratic dependence in $d$ is eventually tight when $n$ grows. More precisely the upper bounds cannot be better than $c(n)d^{2-2/n}{H}^{n-1}$ in general. Note that for affine dimension growth (for affine hypersurfaces of degree $d$, satisfying some extra conditions), the dependence on $d$ is also quadratic by [1], which is already known to be optimal by [3]. Our projective case thus complements the picture of tightness for dimension growth bounds for hypersurfaces.