Adjoint representation of E_8 and del Pezzo surfaces of degree 1

TL;DR

This paper constructs a G-equivariant embedding of a universal torsor over a degree 1 del Pezzo surface into the E_8 adjoint orbit, linking geometric and Lie-theoretic structures.

Contribution

It extends previous work by providing a G-equivariant embedding for degree 1 del Pezzo surfaces into the E_8 adjoint representation.

Findings

Embedding relates roots of G to exceptional curves on X

T-invariant hyperplane sections correspond to exceptional curves

Extends previous results to degree 1 del Pezzo surfaces

Abstract

Let X be a del Pezzo surface of degree 1, and let G be the simple Lie group of type E_8. We construct a locally closed embedding of a universal torsor over X into the G-orbit of the highest weight vector of the adjoint representation. This embedding is equivariant with respect to the action of the Neron-Severi torus T of X identified with a maximal torus of G extended by the group of scalars. The T-invariant hyperplane sections of the torsor defined by the roots of G are inverse images of the 240 exceptional curves on X. This extends the main result of our previous work to del Pezzo surfaces of degree 1.