Abstract
For a smooth projective variety X with exceptional structure sheaf, and {X}^{[2]} the Hilbert scheme of two points on X, we show that the Fourier–Mukai functor {{,mathrm{mathbf {D}},}}^mathrm {b}(X) rightarrow {{,mathrm{mathbf {D}},}}^mathrm {b}({X}^{[2]}) induced by the universal ideal sheaf is fully faithful, provided the dimension of X is at least 2. This fully faithfulness allows us to construct a spectral sequence relating the deformation theories of X and {X}^{[2]} and to show that it degenerates at the second page, giving a Hochschild–Kostant–Rosenberg-type filtration on the Hochschild cohomology of X. These results generalise known results for surfaces due to Krug–Sosna, Fantechi and Hitchin. Finally, as a by-product, we discover the following surprising phenomenon: for a smooth projective variety of dimension at least 3 with exceptional structure sheaf, it is rigid if and only if its Hilbert scheme of two points is rigid. This last fact contrasts drastically to the surface case: non-commutative deformations of a surface contribute to commutative deformations of its Hilbert square.
Highlights
Hilbert schemes of points are one of the most tractable and well-studied moduli spaces of sheaves
For surfaces the situation is well-understood: the deformation theory is described by Fantechi and Hitchin in [11,16], and the derived categories of Hilbert schemes are studied by e.g. Bridgeland–King–Reid, Haiman and Krug–Sosna [5,15,22]
In this paper we focus on the case of n = 2, i.e. that of Hilbert squares, and explain how results from the surface case generalise to higher dimensions
Summary
Hilbert schemes of points are one of the most tractable and well-studied moduli spaces of sheaves. Hitchin’s construction is very geometric, and the observation that started this work is that (1) can be recovered more abstractly using derived categories and Hochschild cohomology For this we use a result due to Krug and Sosna: for a smooth projective surface S for which OS is exceptional (i.e. Hi (S, OS) = 0 for i ≥ 1) the Fourier–Mukai functor. Whose kernel I is the ideal sheaf of the universal family Z → S × S[n], is fully faithful [22, theorem 1.2] It follows that there is an isomorphism. King–Reid–Haiman equivalence [5,15], our proof of Theorem A uses the explicit geometry of X [2], which works in arbitrary dimension It provides an alternative proof of the Krug–Sosna result in the case of 2 points.
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