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Gosset Polytopes in Picard Groups of del Pezzo Surfaces

  • Jae-Hyouk Lee (a1)
Abstract

In this article, we study the correspondence between the geometry of del Pezzo surfaces Sr and the geometry of the r-dimensional Gosset polytopes (r − 4)21. We construct Gosset polytopes (r −4)21 in Pic Sr ꕕ ℚ whose vertices are lines, and we identify divisor classes in Pic Sr corresponding to (a − 1)-simplexes (ar), (r − 1)-simplexes and (r − 1)-crosspolytopes of the polytope (r − 4)21. Then we explain how these classes correspond to skew a-lines(ar), exceptional systems, and rulings, respectively.

As an application, we work on the monoidal transform for lines to study the local geometry of the polytope (r−4)21. And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes 321 and 421, respectively.

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References
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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