New Class of Integrable Maps of the Plane: Manin Transformations with Involution Curves

TL;DR

This paper introduces involution curves in cubic pencils and demonstrates how they can be used to construct integrable plane maps that preserve these pencils, advancing the understanding of integrable systems.

Contribution

It defines involution curves for cubic pencils and shows their application in constructing new integrable maps of the plane.

Findings

Involution curves intersect each cubic pencil curve at exactly one non-base point.

Involution curves enable the construction of integrable maps that preserve cubic pencils.

The approach provides a new method for generating integrable plane transformations.

Abstract

For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to construct integrable maps of the plane which leave invariant a cubic pencil.