Hilbert squares: derived categories and deformations

Abstract

For a smooth projective variety $X$ with exceptional structure sheaf, and ${\mathrm{Hilb}}^{2}X$ the Hilbert scheme of two points on $X$, we show that the Fourier-Mukai functor ${\mathbf{D}}^{\mathrm{b}}(X)\to {\mathbf{D}}^{\mathrm{b}}({\mathrm{Hilb}}^{2}X)$ 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 ${\mathrm{Hilb}}^{2}X$ 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.

arXiv comment28 pages, all comments welcome, added a reference to a recent faithfulness result

References in corpus (9)

• Blowing up non-commutative smooth surfaces
• Hochschild homology and semiorthogonal decompositions
• The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem
• Geometric realizations of quiver algebras
• Does full imply faithful?
• Hochschild cohomology of noncommutative planes and quadrics
• Fourier-Mukai transform of vector bundles on surfaces to Hilbert scheme
• Some ways to reconstruct a sheaf from its tautological image on a Hilbert scheme of points
• Derived categories of noncommutative quadrics and Hilbert squares

Cited by in corpus (6)

• Derived categories of (nested) Hilbert schemes
• Derived categories of flips and cubic hypersurfaces
• Decompositions of moduli spaces of vector bundles and graph potentials
• Polyvector fields for Fano 3-folds
• Vector fields and admissible embeddings for quiver moduli
• Smooth components on special iterated Hilbert schemes