Rational Points and Non-anticanonical Height Functions

A conjecture of Batyrev and Manin predicts the asymptotic behaviourof rational points of bounded height on smooth projective varieties overnumber fields. We prove some new cases of this conjecture for conic bundle surfacesequipped with some non-anticanonical height functions. As a special case,we verify these conjectures for the first time for some smooth cubic surfaces forheight functions associated to certain ample line bundles.