Generalised Manin transformations and QRT maps

TL;DR

This paper generalizes Manin transformations to birational maps preserving quadratic and quartic pencils, demonstrating their measure-preserving properties, symmetries, and relation to QRT maps, thus broadening the scope of integrable plane maps.

Contribution

It introduces explicit birational maps that preserve quadratic and quartic pencils, extending Manin transformations and connecting them to QRT maps through a unifying framework.

Findings

Maps preserve measure and are integrable.

Quadratic pencils admit infinitely many symmetries.

QRT maps are special cases of the quartic generalization.

Abstract

Manin transformations are maps of the plane that preserve a pencil of cubic curves. They are the composition of two involutions. Each involution is constructed in terms of an involution point that is required to be one of the base points of the pencil. We generalise this construction to explicit birational maps of the plane that preserve quadratic resp. certain quartic pencils, and show that they are measure-preserving and hence integrable. In the quartic construction the two involution points are required to be base points of the pencil of multiplicity 2. On the other hand, for the quadratic pencils the involution points can be any two distinct points in the plane (except for base points). We employ Pascal's theorem to show that the maps that preserve a quadratic pencil admit infinitely many symmetries. The full 18-parameter QRT map is obtained as a special instance of the quartic case in a limit where the two involution points go to infinity. We show by construction that each generalised Manin transformation can be brought to QRT form by a fractional affine transformation. We also specify classes of generalised Manin transformations which admit a root.