Gosset polytopes in picard groups of Del Pezzo surfaces

In this article, we study the correspondence between the geometry of del Pezzo surfaces S_r and the geometry of the r-dimensional 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 - 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.