Serre's question on thin sets in projective space
Tijs Buggenhout, Raf Cluckers, Per Salberger, Tim Santens, Floris Vermeulen
TL;DR
The paper resolves Serre's question on rational points of bounded height for projective thin sets of type II in degree $d\ge 4$, uniformly across dimensions and over all global fields. It develops an affine reduction and a weighted determinant method for non-homogeneous polynomials, augmented by sharp quadratic-in-$d$ bounds for points on curves from recent work, and a careful analysis of twisted lines on surfaces. By constructing auxiliary weighted polynomials and counting in skew boxes, the authors obtain bounds $N_{ m cover}(f,B)\ll B^{n}$ for $d\ge 5$ (and refined bounds for $d=4,2,3$), with polynomial dependence on $d$ in the affine setting and no logarithmic factors for $d\ge 5$; these translate to the projective case via a Broberg-type reduction. The results extend to global fields, with analogous affine and projective bounds, and include a detailed treatment of the curve case, yielding sharp height-based counts. Overall, the work advances the arithmetic of thin sets by delivering a uniform, degree-sensitive framework that eliminates $\varepsilon$-losses and logarithmic penalties in higher degrees while broadening applicability to all characteristics and global fields.
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.