• Source: 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).




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, 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
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, vol. 1637, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0094399, ISBN 978-3-540-61795-2, MR 1438547




External links


Weisstein, Eric W. "Burkhardt quartic". MathWorld.

Kata Kunci Pencarian:

• Burkhardt quartic
• Quartic threefold
• Heinrich Burkhardt
• Siegel modular variety
• November 1914
• Hanseatic League
• Coeliac disease
• Capitalism as Religion
• History of group theory