Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Compactification of a Drinfeld period domain over a finite field

Authors: Richard Pink and Simon Schieder
Journal: J. Algebraic Geom. 23 (2014), 201-243
Published electronically: October 17, 2013
MathSciNet review: 3166390
Full-text PDF

Abstract | References | Additional Information

Abstract: We study a certain compactification of the Drinfeld period domain over a finite field which arises naturally in the context of Drinfeld moduli spaces. Its boundary is a disjoint union of period domains of smaller rank, but these are glued together in a way that is dual to how they are glued in the compactification by projective space. This compactification is normal and singular along all boundary strata of codimension  $ \geqslant 2$. We study its geometry from various angles including the projective coordinate ring with its Hilbert function, the cohomology of twisting sheaves, the dualizing sheaf, and give a modular interpretation for it. We construct a natural desingularization which is smooth projective and whose boundary is a divisor with normal crossings. We also study its quotients by certain finite groups.

References [Enhancements On Off] (What's this?)

  • [1] Mauro Beltrametti and Lorenzo Robbiano, Introduction to the theory of weighted projective spaces, Exposition. Math. 4 (1986), no. 2, 111–162. MR 879909
  • [2] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
  • [3] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • [4] Sascha Orlik, Kohomologie von Periodenbereichen über endlichen Körpern, J. Reine Angew. Math. 528 (2000), 201–233 (German, with German summary). MR 1801662, 10.1515/crll.2000.091
  • [5] Pink, R.: Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank. Manuscripta Math. 140 (2013) 333-361.
  • [6] Michael Rapoport, Period domains over finite and local fields, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 361–381. MR 1492528
  • [7] Clarence Wilkerson, A primer on the Dickson invariants, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 421–434. MR 711066, 10.1090/conm/019/711066

Additional Information

Richard Pink
Affiliation: Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland

Simon Schieder
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Received by editor(s): January 26, 2011
Received by editor(s) in revised form: June 7, 2011
Published electronically: October 17, 2013
Additional Notes: The second author was supported by the International Fulbright Science and Technology Award of the U.S. Department of State
Article copyright: © Copyright 2013 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
AMS Website