Counting rational points on smooth quartic and quintic surfaces

Let $X\subseteq {\mathbb{P}}^3$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\mathrm{\prime }}}(B)$ 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 $N_{X^{\mathrm{\prime }}}(B){\ll }_{K,d,\epsilon }{B}^{4\mathrm{/}3+\epsilon }$ for any fixed $\epsilon >0$. This improves an unconditional 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 $X$. More generally, we prove that if $X\subseteq {\mathbb{P}}^n$ is a non-degenerate non-uniruled smooth projective surface defined over $K$, then $N_{X^{\mathrm{\prime }}}(B){\ll }_{K,n,d,\epsilon }{B}^{\frac{n+1}{n}+\epsilon }$.