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.
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).
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Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090.