Gosset Polytopes in Picard Groups of del Pezzo Surfaces

Abstract

Abstract 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 Sr ꕕ 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 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.