Compactification of a Drinfeld period domain over a finite field
Authors:
Richard Pink and Simon Schieder
Journal:
J. Algebraic Geom. 23 (2014), 201-243
DOI:
https://doi.org/10.1090/S1056-3911-2013-00605-0
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
- Mauro Beltrametti and Lorenzo Robbiano, Introduction to the theory of weighted projective spaces, Exposition. Math. 4 (1986), no. 2, 111–162. MR 879909
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Sascha Orlik, Kohomologie von Periodenbereichen über endlichen Körpern, J. Reine Angew. Math. 528 (2000), 201–233 (German, with German summary). MR 1801662, DOI https://doi.org/10.1515/crll.2000.091
- Pink, R.: Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank. Manuscripta Math. 140 (2013) 333–361.
- 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
- 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, DOI https://doi.org/10.1090/conm/019/711066
References
- Mauro Beltrametti and Lorenzo Robbiano, Introduction to the theory of weighted projective spaces, Exposition. Math. 4 (1986), no. 2, 111–162. MR 879909 (88h:14052)
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956 (95h:13020)
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57 \#3116)
- Sascha Orlik, Kohomologie von Periodenbereichen über endlichen Körpern, J. Reine Angew. Math. 528 (2000), 201–233 (German, with German summary). MR 1801662 (2002d:11071), DOI https://doi.org/10.1515/crll.2000.091
- Pink, R.: Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank. Manuscripta Math. 140 (2013) 333–361.
- 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 (98m:14013)
- 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 (85c:55017), DOI https://doi.org/10.1090/conm/019/711066
Additional Information
Richard Pink
Affiliation:
Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland
MR Author ID:
139765
Email:
pink@math.ethz.ch
Simon Schieder
Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email:
schieder@math.harvard.edu
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.