Burkhardt quartic
In mathematics, the Burkhardt quartic is a quartic threefold in 4-dimensional projective space studied by Burkhardt (1890, 1891, 1892), with the maximum possible number of 45 nodes.
Definition
[edit]The equations defining the Burkhardt quartic become simpler if it is embedded in P5 rather than P4. In this case it can be defined by the equations σ1 = σ4 = 0, where σi is the ith elementary symmetric function of the coordinates (x0 : x1 : x2 : x3 : x4 : x5) of P5.
Properties
[edit]The automorphism group of the Burkhardt quartic is the Burkhardt group U4(2) = PSp4(3), a simple group of order 25920, which is isomorphic to a subgroup of index 2 in the Weyl group of E6.
The Burkhardt quartic is rational and furthermore birationally equivalent to a compactification of the Siegel modular variety A2(3).[1]
References
[edit]- ^ Hulek, Klaus; Sankaran, G. K. (2002). "The Geometry of Siegel Modular Varieties". Advanced Studies in Pure Mathematics. 35: 89–156.
- Burkhardt, Heinrich (1890), "Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen Erster Theil", Mathematische Annalen, 36 (3): 371–434, doi:10.1007/BF01206368[permanent dead link]
- Burkhardt, Heinrich (1891), "Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen Zweiter Theil", Mathematische Annalen, 38 (2), Springer: 161–224, doi:10.1007/BF01199251, archived from the original on 2016-03-05, retrieved 2013-09-12
- Burkhardt, Heinrich (1892), "Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen Dritter Theil", Mathematische Annalen, 41 (3): 313–343, doi:10.1007/BF01443416[permanent dead link]
- de Jong, A. J.; Shepherd-Barron, N. I.; Van de Ven, Antonius (1990), "On the Burkhardt quartic", Mathematische Annalen, 286 (1): 309–328, doi:10.1007/BF01453578, ISSN 0025-5831, MR 1032936[permanent dead link]
- Freitag, Eberhard; Salvati Manni, Riccardo (2004), "The Burkhardt group and modular forms", Transformation Groups, 9 (1): 25–45, doi:10.1007/s00031-004-7002-6, ISSN 1083-4362, MR 2130601
- Freitag, Eberhard; Manni, Riccardo Salvati (2006), "Hermitian modular forms and the Burkhardt quartic", Manuscripta Mathematica, 119 (1): 57–59, doi:10.1007/s00229-005-0603-0, ISSN 0025-2611, MR 2194378
- Hunt, Bruce (1996), The geometry of some special arithmetic quotients, Lecture Notes in Mathematics, vol. 1637, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0094399, ISBN 978-3-540-61795-2, MR 1438547