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

Jaume Alonso, Yuri B. Suris

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.

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

Abstract

We find a general framework for the construction of birational involutions on two- and three-dimensional varieties obtained from , , and 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 along conics and along nodal cubic curves, as well as birational involutions of 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.
Paper Structure (26 sections, 3 theorems, 66 equations)

This paper contains 26 sections, 3 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Three scenarios
  3. Scenario 1: Special configuration of nine points in $\mathbb P^2$.
  4. Scenario 2: Special configuration of eight points in $\mathbb P^1\times \mathbb P^1$.
  5. Scenario 3: Special configuration of eight points in $\mathbb P^3$.
  6. General construction of geometric birational involutions
  7. Action of geometric birational involutions on the Picard group
  8. Scheme of the proof for scenarios 1 and 2
  9. Proof for scenario 3
  10. Geometric interpretation of translations in the affine Weyl group
  11. Conclusions
  12. Formulas for the action of geometric involutions on the Picard group
  13. Scenario 1
  14. $\beta=D-E_i$
  15. $\beta=2D-E_1-E_2-E_3-E_4$
  16. ...and 11 more sections

Key Result

Theorem 1

For $\beta\in B$, the birational involution ${\mathcal{I}}_\beta$ induce the following linear involution on the Picard group $I_\beta=({\mathcal{I}}_\beta)_*: {\rm Pic}(X)\to{\rm Pic}(X)$ :

Theorems & Definitions (5)

  • Theorem 1
  • Proposition 1
  • proof
  • Theorem 2
  • proof