Pappus's Theorem in Grassmannian Gr(3,C^n)

TL;DR

This paper explores the geometric and combinatorial structures within Grassmannians, providing a new perspective on Pappus's Theorem by linking it to intersections of quadrics and the discriminantal arrangement.

Contribution

It offers an alternative proof of Pappus's Theorem using Grassmannian geometry and introduces a novel connection with the intersection lattice of discriminantal arrangements.

Findings

Retrieves Pappus's and Hesse configurations as special points in complex Grassmannian.

Provides a new proof of Pappus's Theorem via Grassmannian intersections.

Establishes a link between combinatorial arrangements and algebraic geometry.

Abstract

In this paper we study intersections of quadrics, components of the hypersurface in Grassmannian $Gr(3, \CC^n)$ introduced in \cite{SoSuSi}. This lead to an alternative statement and proof of Pappus's Theorem retrieving Pappus's and Hesse configurations of lines as special points in complex projective Grassmannian. This new connection is obtained through a third purely combinatorial object, the intersection lattice of Discriminantal arrangement.