The action of the Frobenius map on rank $2$ vector bundles in characteristic $2$
Authors:
Y. Laszlo and C. Pauly
Journal:
J. Algebraic Geom. 11 (2002), 219-243
DOI:
https://doi.org/10.1090/S1056-3911-01-00310-1
Published electronically:
November 27, 2001
MathSciNet review:
1874113
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We compute the rational morphism of “inverse image by Frobenius” acting on the coarse moduli space of semi-stable rank $2$ bundles of trivial determinant over an ordinary genus curve in characteristic $2$.
[B]beau A. Beauville: Fibrés de rang 2 sur une courbe, fibrés déterminant et fonctions thêta II, Bull. Soc. Math. France 119 (1991), 259-291
[E]Eke T. Ekedahl: The action of monodromy on torsion points of Jacobians. Arithmetic algebraic geometry (Texel, 1989), 41-49 Birkhäuser Boston, Boston, MA, 1991
[F]fal G. Faltings: Semistable vector bundles on Mumford curves, Inv. Math. 74(2) (1983), 199-212
[G]gie D. Giesecker: Stable vector bundles and the Frobenius morphism, Ann. Sci. Ecole Norm. Sup. (4), 6 (1973), 95-101
[H]hud R.W.H. Hudson: Kummer’s Quartic Surface, Cambridge 1905, modern edition, Cambridge University Press, 1990
[JX]xia K. Joshi, E.Z. Xia: Moduli of Vector Bundles on Curves in Positive Characteristic, Compositio Math. 122 (2000), 315-321
[K]kat N. Katz: Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Inst. Hautes Etudes Sci. Publ. Math. 39 (1970), 175-232
[Ke1]kempf1 G. Kempf: Abelian Integrals, Monograf. Inst. Math. 13, Univ. Nac. Autónoma Mexico, Mexico City (1983)
[Ke2]kempf G. Kempf: Multiplication over Abelian Varieties, Amer. J. Math. 110 (1988), 765-773
[LB]lb H. Lange, C. Birkenhake: Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften 302, Springer-Verlag, Berlin, 1992
[LS]ls H. Lange, U. Stuhler: Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Z. 156 (1) (1977), 73-83
[MR]MehRam V.B. Metha, T.R. Ramadas: Moduli of vector bundles, Frobenius splitting and invariant theory, Ann. of Math. (2) 144 (1996), 269-313
[Mi]mi Y. Miyaoka: The Chern Classes and Kodaira Dimension of a Minimal Variety, Adv. Studies in Pure Math. 10 (1987), 449-476
[Mu1]mum1 D. Mumford: On the equations defining Abelian Varieties. I., Invent. Math. 1 (1966), 287-354
[Mu2]mum2 D. Mumford: Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Bombay, 1970
[NR]NarRam M.S. Narasimhan, S. Ramanan: Moduli of vector bundles on a compact Riemann surface, Ann. of Math. 89 (2) (1969), 14-51
[O]oda T. Oda: Vector bundles on an elliptic curve, Nagoya Math. J. 43 (1971)
[R]Ray M. Raynaud: Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France 110 (1) (1982), 103-125
[Sek]sek T. Sekiguchi: On projective normality of Abelian Varieties. II. J. Math. Soc. Japan 29 (4) (1977), 709-727
[Ses]ses C.S. Seshadri: Geometric reductivity over arbitrary base, Advances in Math. 26 (1977), 225-274
[S]nsb N.I. Shepherd-Barron: Semi-stability and reduction mod $p$, Topology 37 (3) (1998), 659-664
[B]beau A. Beauville: Fibrés de rang 2 sur une courbe, fibrés déterminant et fonctions thêta II, Bull. Soc. Math. France 119 (1991), 259-291
[E]Eke T. Ekedahl: The action of monodromy on torsion points of Jacobians. Arithmetic algebraic geometry (Texel, 1989), 41-49 Birkhäuser Boston, Boston, MA, 1991
[F]fal G. Faltings: Semistable vector bundles on Mumford curves, Inv. Math. 74(2) (1983), 199-212
[G]gie D. Giesecker: Stable vector bundles and the Frobenius morphism, Ann. Sci. Ecole Norm. Sup. (4), 6 (1973), 95-101
[H]hud R.W.H. Hudson: Kummer’s Quartic Surface, Cambridge 1905, modern edition, Cambridge University Press, 1990
[JX]xia K. Joshi, E.Z. Xia: Moduli of Vector Bundles on Curves in Positive Characteristic, Compositio Math. 122 (2000), 315-321
[K]kat N. Katz: Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Inst. Hautes Etudes Sci. Publ. Math. 39 (1970), 175-232
[Ke1]kempf1 G. Kempf: Abelian Integrals, Monograf. Inst. Math. 13, Univ. Nac. Autónoma Mexico, Mexico City (1983)
[Ke2]kempf G. Kempf: Multiplication over Abelian Varieties, Amer. J. Math. 110 (1988), 765-773
[LB]lb H. Lange, C. Birkenhake: Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften 302, Springer-Verlag, Berlin, 1992
[LS]ls H. Lange, U. Stuhler: Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Z. 156 (1) (1977), 73-83
[MR]MehRam V.B. Metha, T.R. Ramadas: Moduli of vector bundles, Frobenius splitting and invariant theory, Ann. of Math. (2) 144 (1996), 269-313
[Mi]mi Y. Miyaoka: The Chern Classes and Kodaira Dimension of a Minimal Variety, Adv. Studies in Pure Math. 10 (1987), 449-476
[Mu1]mum1 D. Mumford: On the equations defining Abelian Varieties. I., Invent. Math. 1 (1966), 287-354
[Mu2]mum2 D. Mumford: Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Bombay, 1970
[NR]NarRam M.S. Narasimhan, S. Ramanan: Moduli of vector bundles on a compact Riemann surface, Ann. of Math. 89 (2) (1969), 14-51
[O]oda T. Oda: Vector bundles on an elliptic curve, Nagoya Math. J. 43 (1971)
[R]Ray M. Raynaud: Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France 110 (1) (1982), 103-125
[Sek]sek T. Sekiguchi: On projective normality of Abelian Varieties. II. J. Math. Soc. Japan 29 (4) (1977), 709-727
[Ses]ses C.S. Seshadri: Geometric reductivity over arbitrary base, Advances in Math. 26 (1977), 225-274
[S]nsb N.I. Shepherd-Barron: Semi-stability and reduction mod $p$, Topology 37 (3) (1998), 659-664
Additional Information
Y. Laszlo
Affiliation:
Institut de Mathématiques de Jussieu, Université Paris VI, Case 82, Analyse Algébrique, UMR 7586, 4, place Jussieu, 75252 Paris Cedex 05 France
Email:
laszlo@math.jussieu.fr
C. Pauly
Affiliation:
Laboratoire J.-A. Dieudonné, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02 France
Email:
pauly@math.unice.fr
Received by editor(s):
May 16, 2000
Published electronically:
November 27, 2001
