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

TL;DR

This paper investigates the bounds on the number of rational points on hypersurfaces, showing that the quadratic dependence on degree in projective dimension growth bounds is ultimately tight as the dimension increases.

Contribution

It demonstrates that the quadratic dependence in the degree for projective dimension growth bounds cannot be improved beyond a certain asymptotic limit for large dimensions.

Findings

Quadratic dependence in degree is tight for large dimensions

Upper bounds cannot be better than c(n)d^{2-2/n} H^{n-1}

Results complement existing bounds for affine hypersurfaces

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.