Gosset Polytopes in Picard groups of del Pezzo Surfaces

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 3_{21} and 4_{21}, respectively.