  Codimension 2 complete intersections in P^N have a natural parameter space
\bar{H}: a projective bundle over a projective space given by the choice of the
lower degree equation and of the higher degree equation up to a multiple of the
first. Motivated by the question of existence of complete families of smooth
complete intersections, we study the birational geometry of \bar{H}. In a first
part, we show that the first contraction of the MMP for \bar{H} always exists
and we describe it. Then, we show that it is possible to run the full MMP for
\bar{H}, and we describe it, in two degenerate cases. As an application, we
prove the existence of complete curves in the punctual Hilbert scheme of
complete intersection subschemes of A^2.
