Towards linear growth for biprojective hypersurfaces of small dimension

Abstract: After removing suitable thin subsets, we count rational points of bounded height on some hypersurfaces of small bidegree in $\mathbb{P}<sup>3\times\mathbb{P}<sup>3$ and in $\mathbb{P}<sup>3\times\mathbb{P}<sup>2$. One of the ingredients is a new estimate for the number of rational points of bounded height on diagonal quadratic surfaces which improves as the coefficients grow.