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

Abstract

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 $ \ell(x,y) $, let $ B_1,B_2 $ be any two distinct points on the line $ \ell(x,y) = -c $, and let $ B_3,B_4 $ be any two distinct points on the line $ \ell(x,y) = c $. Set $ B_0 = \tfrac{1}{2}(B_1+B_3) $ and $ B_5 = \tfrac{1}{2}(B_2+B_4) $; these points lie on the line $ \ell(x,y) = 0 $. Finally, let $ B_\infty $ be the point at infinity on this line. Let $ \mathfrak E $ be the pencil of conics with the base points $ B_1,B_2,B_3,B_4 $. Then the composition of the $ B_\infty $-switch and of the $ B_0 $-switch on the pencil $ \mathfrak E $ is the Kahan discretization of a Hamiltonian vector field $ f = \ell(x,y)\begin{pmatrix}\partial H/\partial y \\ -\partial H/\partial x \end{pmatrix} $ with a quadratic Hamilton function $ H(x,y) $. This birational map $ \Phi_f:\mathbb C P^2\dashrightarrow\mathbb C P^2 $ has three singular points $ B_0,B_2,B_4 $, while the inverse map $ \Phi_f^{-1} $ has three singular points $ B_1,B_3,B_5 $.