Extension groups of tautological sheaves on Hilbert schemes

Krug, Andreas

doi:10.1090/S1056-3911-2014-00655-X

Author: Andreas Krug
Journal: J. Algebraic Geom. 23 (2014), 571-598
DOI: https://doi.org/10.1090/S1056-3911-2014-00655-X
Published electronically: February 25, 2014
MathSciNet review: 3205591
Full-text PDF

Abstract | References | Additional Information

Abstract: We give formulas for the extension groups between tautological sheaves and more generally between tautological objects twisted by natural line bundles on the Hilbert scheme of points on a smooth quasi-projective surface. As a consequence we observe that a tautological object can never be a spherical or $\mathbb {P}^n$-object. We also provide a description of the Yoneda products.

References

Samuel Boissière and Marc A. Nieper-Wißkirchen, Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces, J. Algebra 315 (2007), no. 2, 924–953. MR 2351901, DOI https://doi.org/10.1016/j.jalgebra.2007.04.027
Tom Bridgeland, Alastair King, and Miles Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535–554. MR 1824990, DOI https://doi.org/10.1090/S0894-0347-01-00368-X
Gentiana Danila, Sections du fibré déterminant sur l’espace de modules des faisceaux semi-stables de rang 2 sur le plan projectif, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 5, 1323–1374 (French, with English and French summaries). MR 1800122
Gentiana Danila, Sur la cohomologie d’un fibré tautologique sur le schéma de Hilbert d’une surface, J. Algebraic Geom. 10 (2001), no. 2, 247–280 (French, with English summary). MR 1811556
Gentiana Danila, Sections de la puissance tensorielle du fibré tautologique sur le schéma de Hilbert des points d’une surface, Bull. Lond. Math. Soc. 39 (2007), no. 2, 311–316 (French, with English summary). MR 2323464, DOI https://doi.org/10.1112/blms/bdl035
J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), no. 1, 53–94 (French). MR 999313, DOI https://doi.org/10.1007/BF01850655
John Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521. MR 237496, DOI https://doi.org/10.2307/2373541
Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941–1006. MR 1839919, DOI https://doi.org/10.1090/S0894-0347-01-00373-3
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2244106
Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. MR 2665168
Daniel Huybrechts and Richard Thomas, $\Bbb P$-objects and autoequivalences of derived categories, Math. Res. Lett. 13 (2006), no. 1, 87–98. MR 2200048, DOI https://doi.org/10.4310/MRL.2006.v13.n1.a7
Masaki Kashiwara and Pierre Schapira, Categories and sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006. MR 2182076
A. G. Kuznetsov, Hyperplane sections and derived categories, Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 3, 23–128 (Russian, with Russian summary); English transl., Izv. Math. 70 (2006), no. 3, 447–547. MR 2238172, DOI https://doi.org/10.1070/IM2006v070n03ABEH002318
Manfred Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999), no. 1, 157–207. MR 1681097, DOI https://doi.org/10.1007/s002220050307
Joseph Lipman and Mitsuyasu Hashimoto, Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Mathematics, vol. 1960, Springer-Verlag, Berlin, 2009. MR 2531717
Marc Nieper-Wißkirchen, Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds, World Scientific Publishing Co., Inc., River Edge, NJ, 2004. MR 2110899
Luca Scala, 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. MR 2569613, DOI https://doi.org/10.1215/00127094-2009-050
Luca Scala, Some remarks on tautological sheaves on Hilbert schemes of points on a surface, Geom. Dedicata 139 (2009), 313–329. MR 2481854, DOI https://doi.org/10.1007/s10711-008-9338-x
Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37–108. MR 1831820, DOI https://doi.org/10.1215/S0012-7094-01-10812-0

Additional Information

Andreas Krug
Affiliation: Universität Bonn, Institut für Mathematik
Email: akrug@math.uni-bonn.de

Received by editor(s): February 25, 2012
Received by editor(s) in revised form: March 8, 2013, August 8, 2013, August 19, 2013, and September 17, 2013
Published electronically: February 25, 2014
Article copyright: © Copyright 2014 University Press, Inc.