Gosset Polytopes in Picard groups of del Pezzo Surfaces

2009/05/19 by Jae-Hyouk Lee, Lee, Jae-Hyouk
Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #math.AG #math.CO

paper · pdf · doi:10.48550/arxiv.0905.3025

29 pages. Change the contents from the last version

arxiv created 2010/01/23 · arxiv updated 2010/02/08

Abstract

In this article, we research on the correspondences between the geometry of del Pezzo surfaces S_r and the geometry of Gosset polytopes (r-4)₂₁. We construct Gosset polytopes (r-4)₂₁ in Pic S_r; Q whose vertices are lines, and we identify divisor classes in Pic S_r 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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