  Here we describe the distribution of rational points on the Hilbert scheme of
two points in the projective plane.
  More specifically, we explicitly describe a two-parameter family of height
functions $H_{s, t}$, such that the height function associated to any
projective embedding is equivalent to some $H_{s, t}$, up to multiplication by
a bounded function. For a certain range of the parameters $(s, t)$, we prove an
asymptotic formula for the number of rational points of bounded height, and for
other $(s, t)$ we obtain an upper bound. The proof establishes an equivalence
to a lattice point counting problem, which we solve using the geometry of
numbers.
