Counting rational points on smooth quartic and quintic surfaces

TL;DR

We address counting $K$-rational points of height at most $B$ on smooth degree $d\ge 4$ surfaces $X\subset \mathbb{P}^3$, excluding lines via $N_{X^{\prime}}(B)$. The method cuts $X$ by planes corresponding to points in the dual space and analyzes plane-section curves using a Uniform Faltings bound together with a Rank Hypothesis on abelian varieties to bound point counts on each component. The key contributions are proving $N_{X^{\prime}}(B) \ll_{X,K,\varepsilon} B^{4/3+\varepsilon}$ for $d=4,5$, leveraging height/boundedness arguments and Hilbert-scheme finiteness to control all plane-section components; genus $0$ and $1$ cases are treated with known bounds and torsion constraints. The result improves Salberger's unconditional bounds for $d=4,5$ and extends to any number field; stronger rank hypotheses could remove the $\varepsilon$ and yield $N_{X^{\prime}}(B) \ll_{X,K} B^{4/3}$.

Abstract

Let $X$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\prime}}(B)$ be the number of rational points of $X$ of height at most $B$ that do not lie on lines contained in $X$. Assuming a suitable hypothesis on the size of the rank of abelian varieties, we show that $N_{X^{\prime}}(B)\ll_X B^{4/3+\varepsilon}$ for any fixed $\varepsilon>0$. This improves an unconditional and uniform bound from Salberger for $d=4$ and $d=5$. The proof, based on an argument of Heath-Brown, consists of cutting $X$ by projective planes and using a uniform version of Faltings's Theorem due to Dimitrov, Gao, and Habegger, to bound the number of rational points on the plane sections of $X$.

Theorems & Definitions (6)

• Lemma 1.1: Uniform Faltings's Theorem (mordell-lang)
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof