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The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

Published online by Cambridge University Press:  01 November 2012

Indranil Biswas ,
Ajneet Dhillon  and
Nicole Lemire
Indranil Biswas
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, [email protected]
Ajneet Dhillon
Affiliation:
Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, [email protected]
Nicole Lemire
Affiliation:
Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, [email protected]
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Abstract

We find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

Keywords

Essential dimensionmoduli stackgerbetwisted sheafparabolic bundle
Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 3 , December 2012 , pp. 455 - 488
Copyright
Copyright © ISOPP 2012

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References

BBN01.Balaji, V., Biswas, I. and Nagaraj, D. S.: Principal bundles over projective manifolds with parabolic structure over a divisor, Tohoku Math. Jour. 53 (2001), 337–367.Google Scholar
BD.Biswas, I. and Dey, A.: Brauer group of a moduli space of parabolic vector bundles over a curve, Jour. K-Theory 8 (2011), 437–449.CrossRefGoogle Scholar
BF03.Berhuy, G. and Favi, G.: Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279–330.Google Scholar
BFRV.Brosnan, P., Reichstein, Z. and Vistoli, A.: Essential dimension of moduli of curves and other algebraic stacks (with an appendix by N. Fakhruddin), Jour. Eur. Math. Soc. 13 (2011), 1079–1112.Google Scholar
Bis97.Biswas, I.: Parabolic bundles as orbifold bundles, Duke Math. Jour. 88 (1997), 305–325.Google Scholar
Bor07.Borne, N., Fibrés paraboliques et champs des racines, Int. Math. Res. Not. IMRN, 16 2007, 1073–7928.Google Scholar
BR97.Buhler, J. and Reichstein, Z.: On the essential dimension of a finite group, Compositio Math. 106 (1997), 159–179.CrossRefGoogle Scholar
CKM07.Thelene, J.-L. Colliot, Karpenko, N. A. and Merkurev, A. S.: Rational surfaces and the canonical dimension of the group PGL6 Algebra i Analiz 19 (2007), 159–178.Google Scholar
DL09.Dhillon, A. and Lemire, N.: Upper bounds for the essential dimension of the moduli stack of SLn-bundles over a curve, Transform. Gr. 14 (2009), 747–770.Google Scholar
Har77.Hartshorne, R.: Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977.Google Scholar
HL97.Huybrechts, D. and Lehn, M.: The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig, 1997.Google Scholar
Kar00.Karpenko, N.: On anisotropy of orthogonal involutions, J. Ramanujan Math. Soc. 15 (2000), 1–22.Google Scholar
KM97.Keel, S. and Mori, S.: Quotients by groupoids, Ann. of Math. 145 (1997), 193–213.Google Scholar
Köc05.Köck, B.: Computing the equivariant Euler characteristic of Zariski and étale sheaves on curves, Homology, Homotopy Appl., 7 (2005), 83–98.CrossRefGoogle Scholar
Lie08.Lieblich, M.: Twisted sheaves and the period-index problem, Compos. Math. 144 (2008), 1–31.Google Scholar
LMB00.Laumon, G. and Moret-Bailly, L.: Champs Algébriques, Springer-Verlag, 2000.Google Scholar
Mer03.Merkurjev, A.: Steenrod operations and degree formulas, Jour. Reine Angew. Math. 565 (2003), 13–26.Google Scholar
Mer09.Merkurjev, A. S.: Essential dimension, in Quadratic forms—algebra, arithmetic, and geometry, 299–325, Contemp. Math., vol. 493, Amer. Math. Soc., Providence, RI, 2009.Google Scholar
MS80.Mehta, V. B. and Seshadri, C. S.: Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205–239.Google Scholar
Re1.Reichstein, Z.: Essential dimension, Proceedings of the International Congress of Mathematicians, Volume II, 162–188, Hindustan Book Agency, New Delhi.Google Scholar
Re2.Reichstein, Z.: On the notion of essential dimension for algebraic groups, Transform. Gr. 5 (2000), 265–304.Google Scholar
Tö99.Töen, B.: Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford, K-Theory 18 (1999), 33–76.CrossRefGoogle Scholar
Yo95.Yokogawa, K.: Infinitesimal deformation of parabolic Higgs sheaves, Internat. Jour. Math. 6 (1995), 125–148.CrossRefGoogle Scholar