Title: Sieving rational points on varieties,
Abstract: A sieve for rational points on suitable varieties is developed, together with
applications to counting rational points in thin sets, the number of varieties
in a family which are everywhere locally soluble, and to the notion of friable
rational points with respect to divisors. In the special case of quadrics,
sharper estimates are obtained by developing a version of the Selberg sieve for
rational points.