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.
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.
@cite_0
2083308781
Let @math be a geometrically bounded symplectic manifold, @math be a closed, regular coisotropic submanifold, and @math be a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of @math is bounded below by the sum of the Betti numbers of @math , provided that the Hofer distance between @math and the identity is small enough and the pair @math is non-degenerate. As an application, I prove a presymplectic non-embedding result. A version of the Arnold-Givental conjecture for coisotropic submanifolds is also discussed.