We contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: (a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special geometry of base points ensuring that certain compositions of Manin involutions are integrable maps of low degree (quadratic Cremona maps). In particular, we identify some integrable Kahan discretizations as compositions of Manin involutions for elliptic pencils of higher degree.

中文翻译：

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椭圆铅笔和离散可积系统的Manin对合

我们为由平面双边图生成的离散可积系统的代数几何研究做出了贡献：（a）我们发现椭圆形铅笔的Manin内卷的几何描述，由较高度的曲线组成，双边等效于立方铅笔（索引为1的哈尔芬铅笔） ，以及（b）我们描述了基点的特殊几何形状，以确保Manin对合的某些成分是低阶可积分图（二次克雷莫纳图）。特别是，我们确定了一些可积分的Kahan离散化，作为较高度椭圆铅笔的Manin对合的组成。