Counting rational points with the determinant method

The determinant method gives upper bounds for the number of rational points of bounded height on or near algebraic varieties defined over global fields. There is a real-analytic version of the method due to Bombieri and Pila and a p-adic version due to Heath-Brown. The aim of our talk is to describe a global refinement of the p-adic method and some applications like a uniform bound for non-singular cubic curves which improves upon earlier bounds of Ellenberg-Venkatesh and Heath-Brown.