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.

## Contents

* 1 Definition
* 2 Properties
* 3 References
* 4 External links

## 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

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

* Burkhardt, Heinrich (1891), "Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen Zweiter Theil", Mathematische Annalen (Springer) 38 (2): 161–224, doi:10.1007/BF01199251, http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002252996&L=1, retrieved 2013-09-12
* 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
* 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
* 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

## External links

* Weisstein, Eric W.. "Burkhardt quartic". http://mathworld.wolfram.com/BurkhardtQuartic.html.

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