Manin involutions for elliptic pencils and discrete integrable systems

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.

22 pp, 3 figures

References in corpus (2)

• Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems
• On the singularity structure of Kahan discretizations of a class of quadratic vector fields

Cited by in corpus (4)

• A three-dimensional generalization of QRT maps
• New Class of Integrable Maps of the Plane: Manin Transformations with Involution Curves
• How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6
• Involutions of Halphen Pencils of Index 2 and Discrete Integrable Systems