On the derived category of the Hilbert scheme of points on an Enriques surface

Krug, Andreas; Sosna, Pawel

doi:10.1007/s00029-015-0178-x

Cited by 22 publications

(28 citation statements)

References 35 publications

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“…More recently, autoequivalences of the (bounded) derived categories D b (X [n] ) of the Hilbert schemes were intensively studied; see [Plo07], [Add11], [PS12], [Mea12], [Kru13], [CLS14], [KS14]. In particular, Addington [Add11] defined the notion of a P n -functor as a Fourier-Mukai transform F : D b (M ) → D b (N ) between derived categories of varieties with rightadjoint F R : D b (N ) → D b (M ) and the main property that In [Kru13] it was shown that for every smooth surface X and n ≥ 2 there is a P n−1 -functor H 0,n := H : D b (X) → D b…”

Section: Introductionmentioning

confidence: 99%

$\mathbb{P}$-functor versions of the Nakajima operators

Krug¹

2019

Alg. Geom.

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For a smooth quasi-projective surface X we construct a series of P n−1 -functors H ℓ,n : D b (X × X [ℓ] ) → D b (X [n+ℓ] ) for n > ℓ and n > 1 using the derived McKay correspondence. They can be considered as analogues of the Nakajima operators. The functors also restrict to P n−1 -functors on the generalised Kummer varieties. We also study the induced autoequivalences and obtain, for example, a universal braid relation in the groups of derived autoequivalences of Hilbert squares of K3 surfaces and Kummer fourfolds.given by the Fourier-Mukai transform along the ideal sheaf of the universal family Ξ ⊂ X ×X [n] for X a K3 surface and n ≥ 2. Similarly, for X = A an abelian surface andΞ ⊂ A×K n A the universal family of the generalised Kummer variety K n A ⊂ A [n+1] the functorF n = FM IΞ : D b (A) → D b (K n A) is a P n−1 -functor for n ≥ 2 and a P 1 -functor for n = 1; see [Mea12] and [KM14]. We will refer to these examples of P-functors as the universal ideal functors.An important tool for the investigation of the derived categories of the Hilbert schemes of points on surfaces is the Bridgeland-King-Reid-Haiman equivalence (also known as derived McKay correspondence; see [BKR01] and [Hai01])Sn (X n ) . This provides an identification of D b (X [n] ) with the derived category of equivariant coherent sheaves on the product X n , i.e. of coherent sheaves equipped with a linearisation by the symmetric group S n .

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Section: Introductionmentioning

confidence: 99%

$\mathbb{P}$-functor versions of the Nakajima operators

Krug¹

2019

Alg. Geom.

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“…For example, if D(Y ) carries a (full, strong) exceptional collection one can construct a (full, strong) exceptional collection on D Sn (Y n ). The same holds for semi-orthogonal decompositions and tilting bundles; see [KS15,Sect. 4].…”

Section: Tautological Objects Under the Derived Mckay Correspondencementioning

confidence: 77%

Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles

Krug

2018

Math. Ann.

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We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface.

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“…where (−) ∨ ≔ RHom(−, O X ×X [2] ) is the derived dual. This notation for the functors mimicks that of [21].…”

Section: Fully Faithfulnessmentioning

confidence: 99%

Hilbert squares: derived categories and deformations

Belmans

Raedschelders

2019

Sel. Math. New Ser.

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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 D b (X ) → D 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 byproduct, 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.

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On the derived category of the Hilbert scheme of points on an Enriques surface

Cited by 22 publications

References 35 publications

$\mathbb{P}$-functor versions of the Nakajima operators

$\mathbb{P}$-functor versions of the Nakajima operators

Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles

Hilbert squares: derived categories and deformations

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