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

In projective dimension growth results, one bounds the number of rational points of height at most H on an irreducible hypersurface in 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.