Gosset Polytopes in Picard groups of del Pezzo Surfaces

TL;DR

This paper explores the relationship between the geometry of del Pezzo surfaces and Gosset polytopes, constructing polytope representations in Picard groups and analyzing symmetries induced by classical transformations.

Contribution

It introduces a novel correspondence between divisor classes on del Pezzo surfaces and specific polytopes, revealing geometric and symmetry properties.

Findings

Vertices of Gosset polytopes correspond to lines on del Pezzo surfaces.

Divisor classes relate to simplexes and crosspolytopes within the polytope.

Transformations induce symmetries in the polytopes.

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