Gosset Polytopes in Picard groups of del Pezzo Surfaces
Abstract
In this article, we research on the correspondences between the geometry of del Pezzo surfaces Sr and the geometry of 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-1)-simplexes, (r-1)-simplexes and (r-1)-crosspolytopes of the polytope (r-4)21. Then we explain these classes correspond to skew a-lines, 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 Gieser transformation and Bertini transformation induce a symmetry of polytopes 321 and 421, respectively.
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