Automorphisms of Deitmar schemes, I. Functoriality and 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.

28 pages ; preprint (may 2016)