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
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
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
- 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
- Burkhardt, Heinrich (1891), "Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen Zweiter Theil", Mathematische Annalen, Springer, 38 (2): 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
- 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
- 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, 1637, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0094399, ISBN 978-3-540-61795-2, MR 1438547