Adjoint representation of 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

Abstract

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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 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 .

How to cite

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Serganova, 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 -

References

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  6. D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory, (1994), Springer-Verlag Zbl0797.14004MR1304906
  7. V.V. Serganova, A.N. Skorobogatov, Del Pezzo surfaces and representation theory, Algebra Number Theory 1 (2007), 393-419 Zbl1170.14026MR2368955
  8. V.V. Serganova, A.N. Skorobogatov, On the equations for universal torsors over del Pezzo surfaces, J. Inst. Math. Jussieu 9 (2010), 203-223 Zbl1193.14051MR2576802
  9. B. Sturmfels, Z. Xu, Sagbi Bases of Cox–Nagata Rings, J. Eur. Math. Soc. 12 (2010), 429-459 Zbl1202.14053MR2608947
  10. D. Testa, A. Várilly-Alvarado, M. Velasco, Cox rings of degree one del Pezzo surfaces, Algebra Number Theory 3 (2009), 729-761 Zbl1191.14047MR2579393

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