# Burkhardt quartic

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In mathematics, the **Burkhardt quartic** is a [quartic threefold](/source/Quartic_threefold) in 4-dimensional projective space studied by txt, with the maximum possible number of 45 nodes.

## Definition

The equations defining the Burkhardt quartic become simpler if it is embedded in *P*5 rather than *P*4. In this case it can be defined by the equations σ1 = σ4 = 0, where σ*i* is the *i*th [elementary symmetric function](/source/Elementary_symmetric_function) of the coordinates (*x*0 : *x*1 : *x*2 : *x*3 : *x*4 : *x*5) of *P*5.

## Properties

The automorphism group of the Burkhardt quartic is the Burkhardt group *U*4(2) = PSp4(3), a simple group of order 25920, which is isomorphic to a subgroup of index 2 in the [Weyl group](/source/Weyl_group) of E6.

The Burkhardt quartic is [rational](/source/Rational_variety) and furthermore [birationally equivalent](/source/Birational_geometry#Birational_maps) to a compactification of the [Siegel modular variety](/source/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 (1890), ["Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen Erster Theil"](https://archive.today/20130912062926/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN00225252X&L=1), *Mathematische Annalen*. **36** (3): 371–434, [doi:10.1007/BF01206368](https://doi.org/10.1007/BF01206368), archived from [the original](http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN00225252X&L=1) on September 12, 2013
- Burkhardt, Heinrich (1891), ["Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen Zweiter Theil"](https://web.archive.org/web/20160305080542/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002252996&L=1), *Mathematische Annalen*. **38** (2): 161–224, Springer, [doi:10.1007/BF01199251](https://doi.org/10.1007/BF01199251), archived from [the original](http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002252996&L=1) on 2016-03-05, retrieved 2013-09-12
- Burkhardt, Heinrich (1892), ["Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen Dritter Theil"](https://archive.today/20130912062945/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002253909&L=1), *Mathematische Annalen*. **41** (3): 313–343, [doi:10.1007/BF01443416](https://doi.org/10.1007/BF01443416), archived from [the original](http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002253909&L=1) on September 12, 2013
- de Jong, A. J.; Shepherd-Barron, N. I.; Van de Ven, Antonius (1990), ["On the Burkhardt quartic"](https://archive.today/20130912062911/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002333597&L=1), *[Mathematische Annalen](/source/Mathematische_Annalen)*. **286** (1): 309–328, [doi:10.1007/BF01453578](https://doi.org/10.1007/BF01453578). [ISSN 0025-5831](https://www.worldcat.org/issn/0025-5831). MR 1032936, archived from [the original](http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002333597&L=1) on September 12, 2013
- 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](https://doi.org/10.1007/s00031-004-7002-6). [ISSN 1083-4362](https://www.worldcat.org/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](https://doi.org/10.1007/s00229-005-0603-0). [ISSN 0025-2611](https://www.worldcat.org/issn/0025-2611). MR 2194378
- Hunt, Bruce (1996), *The geometry of some special arithmetic quotients*, Vol. 1637, Lecture Notes in Mathematics, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [doi:10.1007/BFb0094399](https://doi.org/10.1007/BFb0094399). ISBN 978-3-540-61795-2. MR 1438547

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