Serre's question on thin sets in projective space

Abstract

We answer a question of Serre from the 1980s on rational points of bounded height on projective thin sets, in degree at least $4$. For degrees $2$ and $3$ we improve the known bounds in general. The focus is on thin sets of type II, namely corresponding to the images of ramified dominant quasi-finite covers of projective space, as thin sets of type I are already well understood via dimension growth results by the third author in 2002 (published in 2023) by a global variant of Heath-Brown's $p$-adic determinant method. For type II, we obtain a uniform affine variant of Serre's question which implies the projective case and for which the implicit constant is furthermore polynomial in the degree. We are able to avoid logarithmic factors when the degree is at least $5$ and we prove our results over any global field, of any characteristic. A key ingredient for obtaining the affine variant comes from Binyamini-Cluckers-Novikov (2024) and Binyamini-Cluckers-Kato (2025) where a question of the third author was answered by providing bounds, for rational points on irreducible curves, which are quadratic in the degree. A second key ingredient is an adaptation of Salberger's global determinant method to the case of weighted polynomials. The third key ingredient is the design of our affine variant of Serre's question, for weighted polynomials which are not necessarily weighted homogeneous.