Geometry of two- and three-dimensional integrable systems related to affine Weyl groups $W(E_8^{(1)})$ and $W(E_7^{(1)})$

TL;DR

This paper develops a unified framework for constructing birational involutions on algebraic varieties related to affine Weyl groups, revealing new involutions along conics, cubic curves, and cubic surfaces, and analyzing their actions on Picard groups.

Contribution

It introduces a general method for creating birational involutions on varieties derived from projective spaces, expanding classical involutions to new geometric contexts and detailing their algebraic actions.

Findings

Constructed birational involutions along conics and cubic curves.

Derived a formula for involution actions on Picard groups.

Decomposed affine Weyl group elements into involutions.

Abstract

We find a general framework for the construction of birational involutions on two- and three-dimensional varieties obtained from $\mathbb P^2$, $\mathbb P^1\times \mathbb P^1$, and $\mathbb P^3$ by blow-up at nine, respectively eight points. Each such involution is based on a divisor class with a one-dimensional linear system with a generic element of genus zero. Classical Manin involutions represent the simplest particular case. Novel, more sophisticated cases identified here include birational involutions of $\mathbb P^2$ along conics and along nodal cubic curves, as well as birational involutions of $\mathbb P^3$ along quadratic cones and along Cayley nodal cubic surfaces. We prove a general formula for the induced action of geometric birational involutions on the respective Picard group, and give a general result about decomposition of translational elements of the respective affine Weyl group of symmetries into a product of two geometric birational involutions.