Burkhardt quartic

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 σ_i is the ith elementary symmetric function of the coordinates (x₀ : x₁ : x₂ : x₃ : x₄ : x₅) of P⁵.

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).{{cite journal|title=The Geometry of Siegel Modular Varieties |last1=Hulek |first1=Klaus |last2=Sankaran |first2=G. K. |journal=Advanced Studies in Pure Mathematics |volume=35 |year=2002 |pages=89–156}}

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Category:3-folds