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Deformation of Hilbert schemes of points on a surface
Full text linkLet S be a smooth projective surface over the complex numbers; let S-(r) be its r-fold symmetric product and S-[r]] the Hilbert scheme of O-dimensional subschemes of length r.
In case K-S is trivial, the deformation theory of S[PI has been studied by Beauville and Fujiki in order to construct examples of higher-dimensional symplectic manifolds. In that case S-[r] has deformations which are not Hilbert schemes of points on a surface.
We prove that under suitable hypotheses (e.g, if S is of general type) this cannot happen; every (small) deformation of S-(r) and S-[r] is induced naturally by a deformation of S (in particular, all deformations of S-(r) are locally trivial)
Symmetric obstruction theories and Hilbert schemes of points on threefolds
No full textWe introduce the notion of symmetric obstruction theory and study symmetric obstruction theories which are compatible with C*-actions. We prove that the contribution of an isolated fixed point under a C*-action to equivariant Donaldson-Thomas type invariants is +/- 1. As an application, we compute weighted Euler characteristics of all Hilbert schemes of points on any 3-fold. Moreover, we calculate the zero-dimensional Donaldson-Thomas invariants of any projective Calabi-Yau 3-fold. This proves a conjecture of Maulik-Nekrasov-Okounkov
The intrinsic normal cone
No full textLet X be an algebraic stack in the sense of Deligne-Mumford. We construct a purely 0-dimensional algebraic stack over X (in the sense of Artin), the intrinsic normal cone C-x. The notion of (perfect) obstruction theory for X is introduced, and it is shown how to construct, given a perfect obstruction theory for X, a pure-dimensional virtual fundamental class in the Chow group of X. We then prove some properties of such classes, both in the absolute and in the relative context, Via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory, and sometimes a perfect one
Obstruction Calculus for Functors of Artin Rings, I
No full textIn this paper we define and study obstruction theories for morphisms of functors of Artin rings. We prove the existence of a universal obstruction theory, and we give explicit criteria for completeness and for linearity. As applications, we extend several results in the literature, removing the finite-dimensionality of the tangent space and the existence of a vector space of obstructions from the assumptions. © 1998 Academic Press
On the Hilbert scheme of curves in higher-dimensional projective space
No full textIn this paper we prove that, for any n ≥ 3, there exist infinitely many r ∈ N and for each of them a smooth, connected curve Crin ℙrsuch that Crlies on exactly n irreducible components of the Hilbert scheme Hilb(ℙr). This is proven by reducing the problem to an analogous statement for the moduli of surfaces of general type
Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers
No full textBy a recent result of Viehweg, protective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties. In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group GMis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ≥ 2. A Galois cover f : X → y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pa1], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pa1]. In this paper we prove two results about abelian covers: first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X) = G. Copyright © 1997 by Marcel Dekker, Inc
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