  Using the circle method, we count integer points on complete intersections in
biprojective space in boxes of different side length, provided the number of
variables is large enough depending on the degree of the defining equations and
certain loci related to the singular locus. Having established these
asymptotics we deduce asymptotic formulas for rational points on such varieties
with respect to the anticanonical height function. In particular, we establish
a conjecture of Manin for certain smooth hypersurfaces in biprojective space of
sufficiently large dimension.
