Let 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).

Deformation of Hilbert schemes of points on a surface / Fantechi, Barbara. - In: COMPOSITIO MATHEMATICA. - ISSN 0010-437X. - 98:2(1995), pp. 205-217.

Deformation of Hilbert schemes of points on a surface

Fantechi Barbara
1995

Abstract

Let 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).
98
2
205
217
http://www.numdam.org/item/CM_1995__98_2_205_0
Fantechi, Barbara
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/82741
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