Gosset polytopes in picard groups of Del Pezzo surfaces

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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 ⊗ ℚ whose vertices are lines, and we identify divisor classes in Pic Sr corresponding to (a - l)-simplexes (a ≤ r), (r - l)-simplexes and (r - l)-crosspolytopes of the polytope (r-4)21. Then we explain how these classes correspond to skew a-lines(a ≤ r), 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.

Original languageEnglish
Pages (from-to)123-150
Number of pages28
JournalCanadian Journal of Mathematics
Issue number1
StatePublished - Feb 2012


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