Projective Self-dual polygons in higher dimensions

TL;DR

This paper explores the geometry of self-dual polygons in higher-dimensional projective spaces, providing explicit constructions, dimension calculations for specific cases, and conjectures extending known invariance properties of the Pentagram map.

Contribution

It introduces an explicit construction of self-dual polygons in higher dimensions and determines the dimensions of their moduli spaces for certain parameters.

Findings

Explicit construction of self-dual polygons in projective spaces.

Dimension formulas for moduli spaces of self-dual polygons.

A conjecture extending the Pentagram map invariance to higher dimensions.

Abstract

Motivated by a question from V. Arnold about self-dual curves in projective spaces, we study {\cal M}_{m,n,k}: the moduli space of m-self-dual n-gons in {\mathbb P}^k. This paper lays out an explicit construction of self-dual polygons, and for specific cases of n and m, provides the dimension of {\cal M}_{m,n,k}. We include a conjecture about the Pentagram map in higher dimensions that generalizes Clebsch's theorem, which states that every pentagon in \mathbb{RP}^2 is invariant under the Pentagram map.