Discussing Knarr's 2-surface of \(\mathbb{R}^4\) which generates the first single shift plane (Q1591231)

\textit{N. Knarr} [Topologische Differenzenflächenebenen, Diplomarbeit, Kiel (1983)] showed that a 4-dimensional shift plane admitting a 7-dimensional collineation group is isomorphic to the shift plane generated by the planar function \(f:{\mathbb R}^2\to{\mathbb R}^2\) given by \(f(s,t)=(st-{1\over 3}s^3,{1\over 2}t^2-{1\over 12}s^4)\). The derivatives of \(f\) give rise to a topological spread of \({\mathbb R}^4\) and thus to a 4-dimensional translation plane. A long-standing open problem whether a differentiable map \({\mathbb R}^2\to{\mathbb R}^2\) whose derivatives lead to a topological spread are planar (and so generate a shift plane) or whether additional properties are needed. As a step in that direction the authors investigate Knarr's plane algebraically. To begin with the affine part of Knarr's surface \(\Lambda^a_{Kn}\), that is, the graph of \(f\), is investigated. All automorphic affinities of \(\Lambda^a_{Kn}\) are computed and it is shown that \(\Lambda^a_{Kn}\) is equiaffinely homogeneous. Furthermore, \(\Lambda^a_{Kn}\) is convex with respect to 1-dimensional affine subspaces of \({\mathbb R}^4\). The authors then determine the projective closure \(\Lambda_{Kn}\) of \(\Lambda^a_{Kn}\) as an algebraic variety in 4-dimensional projective space \(\Pi_4\). It is shown that this closure is unique and that one has to add the points on the line at infinity \(\ell_{34}=\{(x_0,\ldots,x_4){\mathbb R}\in\Pi_4\mid x_0=x_1=x_2=0\}\). In fact, \(\ell_{34}\) is the only line of \(\Pi_4\) entirely contained in \(\Lambda_{Kn}\). Moreover, Knarr's surface \(\Lambda_{Kn}\) can algebraically be described as the intersection of the cones \(C_3=\{(x_0,\ldots,x_4){\mathbb R}\in\Pi_4\mid -12x_0^3x_4+6x_0^2x_2^2-x_1^4=0\}\) and \(C_4=\{(x_0,\ldots,x_4){\mathbb R}\in\Pi_4\mid -3x_0^2x_3+3x_0x_1x_2-x_1^3=0\}\) and the cubic hypervariety \(V_3=\{(x_0,\ldots,x_4){\mathbb R}\in\Pi_4\mid 4x_0x_2x_4-3x_0x_3^3-4x_1^2x_4+5x_1x_2x_3-2x_2^3=0\}\). Projection from \((0,0,0,0,1){\mathbb R}\) onto the hyperplane \(H=\{(x_0,\ldots,x_4){\mathbb R}\in\Pi_4\mid x_4=0\}\) yields Cayley's surface \(\Phi_{Cay}=H\cap C_3\), which is an algebraic rational ruled surface of degree 3, and allows to translate questions about Knarr's surface to questions about Cayley's surface. Exploiting this correspondence the authors show that there are two classes of planes that carry 1-dimensional algebraic varieties contained in \(\Lambda_{Kn}\), the planes containing \(\ell_{34}\) and planes carrying a parameter conic. Finally, it is shown that each automorphic collineation of \(\Lambda_{Kn}\) fixes the line \(\ell_{34}\) and is the projective extension of an automorphic affinity of \(\Lambda^a_{Kn}\).