Adjoint representation of E₈ and del Pezzo surfaces of degree 1

2010/05/07 by Vera Serganova, Alexei N. Skorobogatov, Serganova, Vera V. +1
Mathematics · #14J26 #22E46 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1005.1272

openalex publication_date 2010/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a del Pezzo surface of degree 1, and let G be the simple Lie group of type E₈. 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.

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