Automorphisms of Deitmar schemes, I. Functoriality and Trees

2016/05/09 by Manuel Merida-Angulo, Merida-Angulo, Manuel, Koen Thas +1
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.1605.02579

28 pages ; preprint (may 2016)

arxiv created 2016/05/09 · arxiv updated 2016/05/10

Abstract

In a recent paper [3], the authors introduced a map F which associates a Deitmar scheme (which is defined over the field with one element, denoted by \mathbbF₁) with any given graph Γ. By base extension, a scheme X_k = F(Γ) ⊗_\mathbbF₁ 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 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 Γ is a finite tree, we will give a precise description of the combinatorial and projective groups, amongst other results.

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