Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper "Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation" by P. van der Kamp et al. [ 5 ], it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form \begin{document}$ \ell(x,y) $\end{document} , let \begin{document}$ B_1,B_2 $\end{document} be any two distinct points on the line \begin{document}$ \ell(x,y) = -c $\end{document} , and let \begin{document}$ B_3,B_4 $\end{document} be any two distinct points on the line \begin{document}$ \ell(x,y) = c $\end{document} . Set \begin{document}$ B_0 = \tfrac{1}{2}(B_1+B_3) $\end{document} and \begin{document}$ B_5 = \tfrac{1}{2}(B_2+B_4) $\end{document} ; these points lie on the line \begin{document}$ \ell(x,y) = 0 $\end{document} . Finally, let \begin{document}$ B_\infty $\end{document} be the point at infinity on this line. Let \begin{document}$ \mathfrak E $\end{document} be the pencil of conics with the base points \begin{document}$ B_1,B_2,B_3,B_4 $\end{document} . Then the composition of the \begin{document}$ B_\infty $\end{document} -switch and of the \begin{document}$ B_0 $\end{document} -switch on the pencil \begin{document}$ \mathfrak E $\end{document} is the Kahan discretization of a Hamiltonian vector field \begin{document}$ f = \ell(x,y)\begin{pmatrix}\partial H/\partial y \\ -\partial H/\partial x \end{pmatrix} $\end{document} with a quadratic Hamilton function \begin{document}$ H(x,y) $\end{document} . This birational map \begin{document}$ \Phi_f:\mathbb C P^2\dashrightarrow\mathbb C P^2 $\end{document} has three singular points \begin{document}$ B_0,B_2,B_4 $\end{document} , while the inverse map \begin{document}$ \Phi_f^{-1} $\end{document} has three singular points \begin{document}$ B_1,B_3,B_5 $\end{document} .