The density of rational lines on hypersurfaces: A bihomogeneous perspective

Abstract

Let F be a non-singular homogeneous polynomial of degree d in n variables. We give an asymptotic formula of the pairs of integer points (x, y) with |x| X and |y| Y which generate a line lying in the hypersurface defined by F , provided that n > 2 d−1 d 4 (d + 1)(d + 2). In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of X and Y .