Graph Varieties in High Dimension

TL;DR

This paper explores the structure of the space of all embeddings of a graph into high-dimensional projective space, revealing its decomposition into smooth subvarieties and providing combinatorial descriptions for specific graph classes.

Contribution

It introduces a detailed decomposition of the picture space into cellules, characterizes irreducible components via a partial order on partitions, and offers formulas for minimal ambient dimensions for graph embeddings.

Findings

Complete descriptions of irreducible components for complete graphs

Descriptions for complete multipartite graphs in any dimension

Formulas for minimum ambient dimension constraints

Abstract

We study the \emph{picture space} $X^d(G)$ of all embeddings of a finite graph $G$ as point-and-line arrangements in an arbitrary-dimensional projective space, continuing previous work on the planar case. The picture space admits a natural decomposition into smooth quasiprojective subvarieties called \emph{cellules}, indexed by partitions of the vertex set of $G$, and the irreducible components of $X^d(G)$ correspond to cellules that are maximal with respect to a partial order on partitions that is in general weaker than refinement. We study both general properties of this partial order and its characterization for specific graphs. Our results include complete combinatorial descriptions of the irreducible components of the picture spaces of complete graphs and complete multipartite graphs, for any ambient dimension $d$. In addition, we give two graph-theoretic formulas for the minimum ambient dimension in which the directions of edges in an embedding of $G$ are mutually constrained.