Duke Mathematical Journal

Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles

Luca Scala

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Let X[n] be the Hilbert scheme of n points on the smooth quasi-projective surface X, and let L[n] be the tautological bundle on X[n] naturally associated to the line bundle L on X. As a corollary of Haiman's results, we express the image Φ(L[n]) of the tautological bundle L[n] for the Bridgeland-King-Reid equivalence Φ:Db(X[n])DSnb(Xn) in terms of a complex CL of Sn-equivariant sheaves in DSnb(Xn) and we characterize the image Φ(L[n]⋅⋅⋅L[n]) in terms of the hyperderived spectral sequence E1p,q associated to the derived k-fold tensor power of the complex CL. The study of the Sn-invariants of this spectral sequence allows us to get the derived direct images of the double tensor power and of the general k-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This easily yields the computation of the cohomology of X[n] with values in L[n]L[n] and ΛkL[n]

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Duke Math. J., Volume 150, Number 2 (2009), 211-267.

First available in Project Euclid: 16 October 2009

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Zentralblatt MATH identifier

Primary: 14C05: Parametrization (Chow and Hilbert schemes) 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
Secondary: 18E30: Derived categories, triangulated categories 20C30: Representations of finite symmetric groups


Scala, Luca. Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles. Duke Math. J. 150 (2009), no. 2, 211--267. doi:10.1215/00127094-2009-050. https://projecteuclid.org/euclid.dmj/1255699340

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