Serre's dimension growth conjecture

Let $X\subseteq\mathbb{P}_{\mathbb{Q}}^{M-1}$ be an irreducible projective variety of degree at least two defined over $\mathbb{Q}$. Let $N_X(B)$ be the number of rational points on $X$ of naive height at most $B$. Serre's conjecture. There is a constant $c>0$ such that

$$N_X(B)\ll_X B^{\dim X}(\log B)^c.$$

This is the original projective-variety formulation of dimension growth. The source notes that Salberger proved a version with $B^\epsilon$ in place of $(\log B)^c$, while sharper bounds are available under additional hypotheses.