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.
@cite_25: Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.
Unlike the varieties @math , @math is always log Fano ( @cite_0 Theorem 2.5). This immediately implies that it is a Mori dream space by @cite_25 , answering the analogue of Question for @math . Since @math is of Picard rank @math , it is possible to run its MMP in two directions. One of these is trivial: we get a contraction, the Hilbert-Chow morphism. As for @math , it is the other one that is interesting to describe.