# Adjoint representation of ${\text{E}}_{8}$ and del Pezzo surfaces of degree $1$

Vera V. Serganova^{[1]}; Alexei N. Skorobogatov^{[2]}

- [1] University of California Department of Mathematics Berkeley, CA, 94720-3840 (USA)
- [2] Imperial College London Department of Mathematics South Kensington Campus SW7 2BZ England, (U.K.) Institute for the Information Transmission Problems Russian Academy of Sciences 19 Bolshoi Karetnyi Moscow, 127994 (Russia)

Annales de l’institut Fourier (2011)

- Volume: 61, Issue: 6, page 2337-2360
- ISSN: 0373-0956

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topSerganova, Vera V., and Skorobogatov, Alexei N.. "Adjoint representation of $\text{\upshape E}_8$ and del Pezzo surfaces of degree $1$." Annales de l’institut Fourier 61.6 (2011): 2337-2360. <http://eudml.org/doc/219764>.

@article{Serganova2011,

abstract = {Let $X$ be a del Pezzo surface of degree $1$, and let $G$ be the simple Lie group of type $\text\{\upshape E\}_8$. 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 Néron-Severi torus $T$ of $X$ identified with a maximal torus of $G$ extended by the group of scalars. Moreover, the $T$-invariant hyperplane sections of the torsor defined by the roots of $G$ are the inverse images of the $240$ exceptional curves on $X$.},

affiliation = {University of California Department of Mathematics Berkeley, CA, 94720-3840 (USA); Imperial College London Department of Mathematics South Kensington Campus SW7 2BZ England, (U.K.) Institute for the Information Transmission Problems Russian Academy of Sciences 19 Bolshoi Karetnyi Moscow, 127994 (Russia)},

author = {Serganova, Vera V., Skorobogatov, Alexei N.},

journal = {Annales de l’institut Fourier},

keywords = {Universal torsors; del Pezzo surfaces; Lie groups; homogeneous spaces; Del Pezzo surface; highest weight vector; orbit; universal torsor; simple Lie group},

language = {eng},

number = {6},

pages = {2337-2360},

publisher = {Association des Annales de l’institut Fourier},

title = {Adjoint representation of $\text\{\upshape E\}_8$ and del Pezzo surfaces of degree $1$},

url = {http://eudml.org/doc/219764},

volume = {61},

year = {2011},

}

TY - JOUR

AU - Serganova, Vera V.

AU - Skorobogatov, Alexei N.

TI - Adjoint representation of $\text{\upshape E}_8$ and del Pezzo surfaces of degree $1$

JO - Annales de l’institut Fourier

PY - 2011

PB - Association des Annales de l’institut Fourier

VL - 61

IS - 6

SP - 2337

EP - 2360

AB - Let $X$ be a del Pezzo surface of degree $1$, and let $G$ be the simple Lie group of type $\text{\upshape E}_8$. 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 Néron-Severi torus $T$ of $X$ identified with a maximal torus of $G$ extended by the group of scalars. Moreover, the $T$-invariant hyperplane sections of the torsor defined by the roots of $G$ are the inverse images of the $240$ exceptional curves on $X$.

LA - eng

KW - Universal torsors; del Pezzo surfaces; Lie groups; homogeneous spaces; Del Pezzo surface; highest weight vector; orbit; universal torsor; simple Lie group

UR - http://eudml.org/doc/219764

ER -

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