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_27: We compute the cone of effective divisors on the Hilbert scheme of points in the projective plane. We show the sections of many stable vector bundles satisfy a natural interpolation condition, and that these bundles always give rise to the edge of the effective cone of the Hilbert scheme. To do this, we give a generalization of Gaeta's theorem on the resolution of the ideal sheaf of a general collection of points in the plane. This resolution has a natural interpretation in terms of Bridgeland stability, and we observe that ideal sheaves of collections of points are destabilized by exceptional bundles. By studying the Bridgeland stability of exceptional bundles, we also show that our computation of the effective cone of the Hilbert scheme is consistent with a conjecture which predicts a correspondence between Mori and Bridgeland walls for the Hilbert scheme.
@cite_7: The Hilbert scheme of n points in the projective plane parameterizes degree n zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert scheme. By studying interpolation, restriction, and stability properties of certain vector bundles on the plane we fully determine these cones for just over three fourths of all values of n. A general Steiner bundle on P^N is a vector bundle admitting a linear resolution of a particular form. We complete the classification of slopes of semistable Steiner bundles on P^N by showing every admissible slope is realized by a bundle which restricts to a semistable bundle on a rational curve. The proof involves a basic question about multiplication of polynomials on P^1 which is interesting in its own right.
The analogue of Question is much more complicated in the case of @math . Indeed, the non-trivial boundary of @math is difficult to describe: it depends on @math in a complicated and interesting fashion ( @cite_7 , @cite_27 Theorem 1.4 and Table 1).