Autoduality of compactified Jacobians for curves with plane singularities
Author:
Dima Arinkin
Journal:
J. Algebraic Geom. 22 (2013), 363-388
DOI:
https://doi.org/10.1090/S1056-3911-2012-00596-7
Published electronically:
September 27, 2012
MathSciNet review:
3019453
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Abstract |
References |
Additional Information
Abstract: Let $C$ be an integral projective curve with at most planar singularities. Consider its Jacobian $J$ and the compactified Jacobian $\overline {J}$. We construct a flat family $\overline {P}$ of Cohen-Macaulay sheaves on $\overline {J}$ parametrized by $\overline {J}$; its restriction to $J\times \overline {J}$ is the Poincaré line bundle. We prove that the Fourier-Mukai transform given by $\overline {P}$ is an auto-equivalence of the derived category of $\overline {J}$.
References
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References
- A. Altman, A. Iarrobino, and S. Kleiman, Irreducibility of the compactified Jacobian, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 1–12. MR 0498546 (58:16650)
- A. Altman and S. Kleiman, Compactifying the Jacobian, Bull. Amer. Math. Soc. 82 (1976), no. 6, 947–949. MR 0429908 (55:2917)
- ---, Compactifying the Picard scheme. II, Amer. J. Math. 101 (1979), no. 1, 10–41. MR 527824 (81f:14025b)
- ---, Compactifying the Picard scheme, Adv. in Math. 35 (1980), no. 1, 50–112. MR 555258 (81f:14025a)
- D. Arinkin, Cohomology of line bundles on compactified Jacobians, Electronic preprint arXiv:0705.0190.
- D. Arinkin and R. Bezrukavnikov, Perverse coherent sheaves, Moscow Math. J. 10 (2010), no. 1, 3–29. MR 2668828 (2011g:14040)
- V. Baranovsky, The variety of pairs of commuting nilpotent matrices is irreducible, Transform. Groups 6 (2001), no. 1, 3–8. MR 1825165 (2003d:14058)
- J. Briançon, Description de $H\textrm {ilb}^ {n}C\{x,y\}$, Invent. Math. 41 (1977), no. 1, 45–89. MR 0457432 (56:15637)
- I. Burban and B. Kreussler, Fourier-Mukai transforms and semi-stable sheaves on nodal Weierstraß cubics, J. Reine Angew. Math. 584 (2005), 45–82. MR 2155085 (2006d:14016)
- R. Donagi and D. Gaitsgory, The gerbe of Higgs bundles, Transform. Groups 7 (2002), no. 2, 109–153. MR 1903115 (2003e:14034)
- R. Donagi and T. Pantev, Langlands duality for Hitchin systems, Electronic preprint arXiv:math/0604617.
- V. Drinfeld, Two-dimensional $l$-adic representations of the fundamental group of a curve over a finite field and automorphic forms on $\textrm {GL}(2)$, Amer. J. Math. 105 (1983), no. 1, 85–114. MR 692107 (84i:12011)
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- E. Esteves and S. Kleiman, The compactified Picard scheme of the compactified Jacobian, Adv. Math. 198 (2005), no. 2, 484–503. MR 2183386 (2006m:14055)
- J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math 90 (1968), 511–521. MR 0237496 (38:5778)
- E. Frenkel, D. Gaitsgory, and K. Vilonen, On the geometric Langlands conjecture, J. Amer. Math. Soc. 15 (2002), no. 2, 367–417. MR 1887638 (2003a:11145)
- D. Gaitsgory, On a vanishing conjecture appearing in the geometric Langlands correspondence, Ann. of Math. (2) 160 (2004), no. 2, 617–682. MR 2123934 (2006k:11223)
- V. Ginzburg, Isospectral commuting variety and the Harish-Chandra $D$-module, Electronic preprint arXiv:1002.3311.
- I. Grojnowski, Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), no. 2, 275–291. MR 1386846 (97f:14041)
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- M. Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941–1006. MR 1839919 (2002c:14008)
- A. Iarrobino, Punctual Hilbert schemes, Mem. Amer. Math. Soc. 10 (1977), no. 188, viii+112. MR 0485867 (58:5667)
- G. Laumon, Faisceaux automorphes pour GL(n): la premiere construction de Drinfeld, Electronic preprint arXiv:alg-geom/9511004.
- ---, Correspondance de Langlands géométrique pour les corps de fonctions, Duke Math. J. 54 (1987), no. 2, 309–359. MR 899400 (88g:11086)
- S. Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175. MR 607081 (82f:14036)
- D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970. MR 0282985 (44:219)
- H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999. MR 1711344 (2001b:14007)
- J. Sawon, Twisted Fourier-Mukai transforms for holomorphic symplectic four-folds, Adv. Math. 218 (2008), no. 3, 828–864. MR 2414323 (2009g:14046)
- E. Tschirnhaus, Methodus auferendi omnes terminos intermedios ex data equatione, Acta Eruditorum (May 1683), 204–207.
Additional Information
Dima Arinkin
Affiliation:
Department of mathematics, University of North Carolina, Chapel Hill, North Carolina
Address at time of publication:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53701
Email:
arinkin@email.unc.edu
Received by editor(s):
August 7, 2010
Received by editor(s) in revised form:
February 27, 2011
Published electronically:
September 27, 2012
Additional Notes:
Supported in part by the Alfred P. Sloan Foundation under the Sloan Research Fellowship program.
