Counting rational points on projective varieties

Abstract 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 P n 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 ) = O d,n, ε ( B dim W +ε ) of this conjecture. We also use our global determinant method to improve upon previous estimates for quasi‐projective surfaces. If, for example, is the complement of the lines on a non‐singular surface X ⊂ P 3 of degree d , then we show that . For surfaces defined by forms with non‐zero coefficients, then we use a new geometric result for Fermat surfaces to show that for B ≥ e .