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.
Here is another difference between @math and @math . The trivial contraction of @math is the fibration over @math , that associates to @math its degree @math equation @math , and the non-trivial contraction that starts the MMP for @math is closely related to a Hilbert-Chow morphism (see for instance the proof Lemma ). On the contrary, the trivial contraction of @math is the Hilbert-Chow morphism and the non-trivial contraction that starts the MMP for @math is given by considering the degree @math equations of a length @math subscheme ( @cite_0 Proposition 3.1).