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Burkhardt quartic

From Wikipedia, the free encyclopedia

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

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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

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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

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  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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