Gosset polytopes in Picard groups of del Pezzo surfaces
The correspondence between the geometry of a del Pezzo surface \(S_r\) and the geometry of \(r\)-dimensional Gosset polytopes \((r-4)_{21}\) is studied. The Gosset polytopes are in \({\text{ Pic}}~S_r\otimes{\mathbb Q}\) whose vertices are lines. It is shown that the divisor classes in \({\text{ Pic}}~ S_r\) correspond to skew lines, exceptional systems, and rulings. As an application, the monoidal transform for lines is used in order to study the local geometry of the polytope \((r-4)_{21}\). Furthermore, it is shown that the Gieser transform and the Bertini transform induce a symmetry of the polytopes \(3_{21}\) and \(4_{21}\).
scientific article; zbMATH DE number 6023029