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

• * Corresponding author: Yuri B. Suris
• 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 $B_1,B_3,B_5$.

Mathematics Subject Classification: Primary: 65P10, 37M15; Secondary: 37J35, 14E05.

 Citation:

• Figure 1.  The black curve is the conic $C = 0$, the red lines represent the reducible conic $D = 0$. The finite base points are $B_1, \ldots, B_4$. The points $B_0$, $B_5$ are singular points of the Kanan map $\Phi_f$, resp. of $\Phi_f^{-1}$, for the following data: $\ell(x,y) = 3x-y$, $H(x,y) = -x^2+(2/5)xy+(1/2)y^2-4x+4y$

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