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 t ...
of E6.
The Burkhardt quartic is
rational
Rationality is the quality of being guided by or based on reason. 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 ...
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 ''A
2(3)''.
References
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External links
*{{mathworld, urlname=BurkhardtQuartic, title=Burkhardt quartic
3-folds