Automorphisms of Deitmar schemes, I. Functoriality and Trees

TL;DR

This paper studies automorphism groups of schemes derived from graphs over fields, showing functorial properties and providing detailed descriptions for schemes from finite trees, advancing understanding of their symmetries.

Contribution

It proves that the graph-to-scheme map is functorial and characterizes automorphism groups for schemes from finite trees, linking combinatorial and geometric symmetries.

Findings

All graph-to-scheme mappings are functors.

Automorphism groups are classified as combinatorial, topological, and scheme-theoretic.

Explicit descriptions of automorphism groups for schemes from finite trees.

Abstract

In a recent paper [3], the authors introduced a map $\mathcal{F}$ which associates a Deitmar scheme (which is defined over the field with one element, denoted by $\mathbb{F}_1$) with any given graph $\Gamma$. By base extension, a scheme $\mathcal{X}_k = \mathcal{F}(\Gamma) \otimes_{\mathbb{F}_1} k$ over any field $k$ arises. In the present paper, we will show that all these mappings are functors, and we will use this fact to study automorphism groups of the schemes $\mathcal{X}_k$. Several automorphism groups are considered: combinatorial, topological, and scheme-theoretic groups, and also groups induced by automorphisms of the ambient projective space. When $\Gamma$ is a finite tree, we will give a precise description of the combinatorial and projective groups, amongst other results.