Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let M r,Lss denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(M r,Lss) = ℤ, identify the ample generator, and deduce that M r,Lss is locally factorial. In characteristic zero, this has already been proved by Drézet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant Theory in positive characteristic.
@article{bwmeta1.element.doi-10_2478_s11533-012-0064-0,
author = {Norbert Hoffmann},
title = {The Picard group of a coarse moduli space of vector bundles in positive characteristic},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1306-1313},
zbl = {1282.14060},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0064-0}
}
Norbert Hoffmann. The Picard group of a coarse moduli space of vector bundles in positive characteristic. Open Mathematics, Tome 10 (2012) pp. 1306-1313. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0064-0/
[1] Beauville A., Laszlo Y., Conformal blocks and generalized theta functions, Commun. Math. Phys., 1994, 164(2), 385–419 http://dx.doi.org/10.1007/BF02101707 | Zbl 0815.14015
[2] Bhosle U.N., Moduli of vector bundles in characteristic 2, Math. Nachr., 2003, 254/255, 11–26 http://dx.doi.org/10.1002/mana.200310049 | Zbl 1056.14044
[3] Biswas I., Hoffmann N., The line bundles on moduli stacks of principal bundles on a curve, Doc. Math., 2010, 15, 35–72 | Zbl 1193.14009
[4] Biswas I., Hoffmann N., Poincaré families of G-bundles on a curve, Math. Ann., 2012, 352(1), 133–154 http://dx.doi.org/10.1007/s00208-010-0628-x | Zbl 1253.14012
[5] Drezet J.-M., Narasimhan M.S., Groupe de Picard des variétés de modules de fibrés semi-stable sur les courbes algébriques, Invent. Math., 1989, 97(1), 53–94 http://dx.doi.org/10.1007/BF01850655 | Zbl 0689.14012
[6] Faltings G., Stable G-bundles and projective connections, J. Algebraic Geom., 1993, 2(3), 507–568 | Zbl 0790.14019
[7] Faltings G., Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. (JEMS), 2003, 5(1), 41–68 http://dx.doi.org/10.1007/s10097-002-0045-x | Zbl 1020.14002
[8] Grothendieck A., Éléments de Géométrie Algébrique. III. Étude Cohomologique des Faisceaux Cohérents. I, II, Inst. Hautes Études Sci. Publ. Math., 11, 17, Presses Universitaires de France, Paris, 1961, 1963
[9] Hoffmann N., Moduli stacks of vector bundles on curves and the King-Schofield rationality proof, In: Cohomological and Geometric Approaches to Rationality Problems, Progr. Math., 282, Birkhäuser, Boston, 2010, 133–148 http://dx.doi.org/10.1007/978-0-8176-4934-0_5 | Zbl 1203.14038
[10] Huybrechts D., Lehn M., The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge, 2010 http://dx.doi.org/10.1017/CBO9780511711985 | Zbl 1206.14027
[11] Joshi K., Mehta V.B., On the Picard group of moduli spaces, preprint available at http://arxiv.org/abs/1005.3007
[12] Knudsen F.F., Mumford D., The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand., 1976, 39(1), 19–55 | Zbl 0343.14008
[13] Mumford D., Fogarty J., Kirwan F., Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb., 34, Springer, Berlin, 1994 http://dx.doi.org/10.1007/978-3-642-57916-5 | Zbl 0797.14004
[14] Narasimhan M.S., Ramanan S., Moduli of vector bundles on a compact Riemann surface, Ann. Math., 1969, 89, 14–51 http://dx.doi.org/10.2307/1970807 | Zbl 0186.54902
[15] Osserman B., The generalized Verschiebung map for curves of genus 2, Math. Ann., 2006, 336(4), 963–986 http://dx.doi.org/10.1007/s00208-006-0026-6 | Zbl 1111.14031
[16] Seshadri C.S., Fibrés Vectoriels sur les Courbes Algébriques, Astérisque, 96, Société Mathématique de France, Paris, 1982 | Zbl 0517.14008
[17] Seshadri C.S., Vector bundles on curves, In: Linear Algebraic Groups and their Representations, Los Angeles, March 25–28, 1992, Contemp. Math., 153, American Mathematical Society, Providence, 1993, 163–200 http://dx.doi.org/10.1090/conm/153/01312 | Zbl 0799.14013
[18] Venkata Balaji T.E., Mehta V.B., Singularities of moduli spaces of vector bundles over curves in characteristic 0 and p, Michigan Math. J., 2008, 57, 37–42 http://dx.doi.org/10.1307/mmj/1220879395 | Zbl 1181.14037