Local asymptotic distribution of rational points

The Batyrev-Manin-Peyre principle predicts some uniform distribution of rational points on rational varieties. We propose a local analogue, whose aim is to describe the local behavior neglected by the global consideration.A heuristic interpretation of the main term valid for all known example is proposed in spirit of the geometric Batyrev-Manin principle together with a conjecture of McKinnon. We give a number of illustrations.