On the birational geometry of parameter spaces for complete intersections

Codimension 2 complete intersections in P^N have a natural parameter space that is 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. In this talk, we will be interested in the birational geometry of this variety: what is its ample cone? Its effective cone? Is it a Mori dream space? We will give geometric motivations for these questions, and answers in particular cases. As an application, we will construct complete curves in the Hilbert schemes of punctual complete intersections in A^2.