Configuration of lines in del Pezzo surfaces with Gosset Polytopes

TL;DR

This paper explores the configurations of lines in del Pezzo surfaces using Gosset polytopes, introducing new combinatorial systems and relating geometric configurations to well-known combinatorial structures.

Contribution

It introduces the k-Steiner system and cornered simplexes, providing a new characterization of line configurations in del Pezzo surfaces via Gosset polytopes.

Findings

Configurations of inscribed simplexes relate to rulings and skew lines in del Pezzo surfaces.

Seven lines in a 6-simplex form a Fano plane.

Higher-dimensional simplexes exist in the Picard group of the degree 1 del Pezzo surface.

Abstract

In this article, we study the divisor classes of del Pezzo surfaces, which are written as the sum of distinct lines with fixed intersection according to the inscribed simplexes and crosspolytopes in Gosset polytopes. We introduce the k-Steiner system and cornered simplexes, and characterize the configurations of inscribed m(<4)-simplexes with them. Higher dimensional inscribed m(3<m)-simplexes exist in 4_{21} in the Picard group of del Pezzo surface S_{8} of degree 1.The configurations of 4- and 7-simplexes are related to rulings in S_{8}. And the configurations of 5- and 6-simplexes correspond the skew 3-lines and skew 7-lines in S_{8}. In particular, the seven lines in a 6-simplex produce a Fano plane. We also study the inscribed crosspolytopes and hypercubes in the Gosset polytopes.