Discrete line complexes and integrable evolution of minors

TL;DR

This paper explores the algebraic and geometric properties of discrete integrable line complexes in complex projective 3-space, linking classical projective geometry with modern integrable systems theory.

Contribution

It introduces a novel algebraic and geometric framework for discrete integrable line complexes based on the Plücker correspondence and Desargues' theorem.

Findings

Discrete integrable systems encode properties of line complexes.

Existence of integrable line complexes is guaranteed by classical projective geometry.

Characterization via correlations of CP^3 is established.

Abstract

Based on the classical Pl\"ucker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in $CP^3$. Algebraically, these are encoded in a discrete integrable system which appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the existence of these integrable line complexes is shown to be guaranteed by Desargues' classical theorem of projective geometry. A remarkable characterisation in terms of correlations of $CP^3$ is also recorded.