Burkhardt quartic

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

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.

Properties

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]