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.
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.
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.
References
- 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
- 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