  A rational distance set in the plane is a point set which has the property
that all pairwise distances between its points are rational.
  Erd\H os and Ulam conjectured in 1945 that there is no dense rational
distance set in the plane.
  In this paper we associate an algebraic surface in $\mathbb{P}^3$, that we
call a distance surface, to any finite rational distance set in the plane.
Under a mild condition, we prove that a distance surface is always a surface of
general type. From this, we deduce that the Bombieri-Lang conjecture in
arithmetic algebraic geometry (restricted to the classes of surfaces) implies
an answer to the Erd\H os-Ulam problem.
  Combined with the results of Solymosi and de Zeeuw, our proofs lead to the
following stronger statement: for $S$ a rational distance set with infinitely
many points, we have
  Either, all but at most four points of $S$ are on a line,
  Or, all but at most three points of $S$ are on a circle.
