In mathematics, the Burkhardt quartic is a
quartic threefold in 4-dimensional projective space studied by ,
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 σ
1 = σ
4 = 0, where σ
''i'' is the ''i''th
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) = PSp
4(3), a simple group of order 25920, which is isomorphic to a subgroup of index 2 in the
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
of E6.
The Burkhardt quartic is
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
and furthermore
birationally equivalent
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...
to a compactification of the
Siegel modular variety
In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally pola ...
''A
2(3)''.
References
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External links
*{{mathworld, urlname=BurkhardtQuartic, title=Burkhardt quartic
3-folds