Codimension 2 complete intersections in P^N have a natural parameter space 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 H . In a first part, we show that the first contraction of the MMP for H always exists and we describe it. Then, we show that it is possible to run the full MMP for 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.
@cite_0: In this paper, we study the birational geometry of the Hilbert scheme of n points on P^2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n at most 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions.
In @cite_0 , Arcara, Bertram, Coskun and Huizenga study the birational geometry of the punctual Hilbert schemes @math of length @math subschemes of @math . This is very related to the case @math of Question and (i.e. to Theorem ) because, in this case, @math is a locally closed subscheme of @math . Let us describe the similarities and differences between these two situations.