Counting rational points on projective varieties

Journal article, 2023

We develop a global version of Heath-Brown's p-adic determinant method to study the asymptotic behaviour of the number N(W; B) of rational points of height at most B on certain subvarieties W of Pn defined over Q. The most important application is a proof of the dimension growth conjecture of Heath-Brown and Serre for all integral projective varieties of degree d ≥ 2 over Q. For projective varieties of degree d ≥ 4, we prove a uniform version N(W; B) = Od,n,ε(BdimW+ε) of this conjecture. We also use our global determinant method to improve upon previous estimates for quasi-projective surfaces. If, for example, (Formula presented.) is the complement of the lines on a non-singular surface X ⊂ P3 of degree d, then we show that (Formula presented.). For surfaces defined by forms (Formula presented.) with non-zero coefficients, then we use a new geometric result for Fermat surfaces to show that (Formula presented.) for B ≥ e.