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 σ_i 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^{[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

External links

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

[1] Hulek, Klaus; Sankaran, G. K. (2002). "The Geometry of Siegel Modular Varieties". Advanced Studies in Pure Mathematics. 35: 89–156.

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