Manin involutions for elliptic pencils and discrete integrable systems

TL;DR

This paper explores the geometric properties of Manin involutions in elliptic pencils and their role in generating integrable discrete systems, linking algebraic geometry with integrability conditions.

Contribution

It provides a geometric description of Manin involutions for higher degree elliptic pencils and characterizes conditions for their compositions to produce integrable maps.

Findings

Identified geometric conditions for base points leading to integrable maps.

Connected certain Kahan discretizations to compositions of Manin involutions.

Described Manin involutions for elliptic pencils of higher degree.

Abstract

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.