Automorphisms of Deitmar constructible sets, I. Functoriality and trees

Abstract

In a recent paper (Mérida-Angulo and Thas, 2017), the authors introduced a map F which associates a Deitmar constructible set (which is defined over the field with one element, denoted by F1) with any given loose graph Γ. By base extension, a constructible set Xk=F(Γ)⊗F1k 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 constructible sets Xk. Several automorphism groups are considered: combinatorial, topological and also groups induced by automorphisms of the ambient projective space. When Γ is a finite tree, we will give a precise description of the combinatorial and projective groups, amongst other results.