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 S_r ? Q whose vertices are lines, and we identify divisor classes in Pic Sr corresponding to (a - 1)-simplexes (a ≤ r), (r - 1)-simplexes and (r - 1)-crosspolytopes of the polytope (r - 4)_(21). Then we explain how these classes correspond to skew α-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 3_(21) and 4_(21), respectively.
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