Combinatorial description of normal toric schemes over valuation rings of rank one

Publisher: Universitaet Tuebingen.Normal toric varieties over a field can be described by combinatorial data, so called rational fans. Mumford did extend this description to normal toric schemes of finite type defined over a discrete valuation ring. We present a generalization of these results for normal toric schemes of finite type defined over a non-necessarily discrete valuation ring of rank one. We start with the affine case by giving a classification in terms of some admissible cones. For the non-affine case the main step is to generalize Sumihiro's theorem for normal toric varieties to this context. Since our schemes are non-noetherian in general, this is done by applying intersection theory with divisors on admissible formal schemes over rank one valuation rings. Finally, using the orbit-cone correspondence for toric varieties over a valuation ring, proved by Gubler, we get that the set of isomorphism classes of normal toric schemes of finite type over a valuation ring is in bijection with the set of fans whose cones are those obtained in the affine case.Introduction- Notation- 1 Toric schemes over valuation rings - 1.1 Affine toric schemes associated to polyhedra - 2 Tropicalization - 1.3 Toric schemes associated to fans - 1.4 Projective toric schemes - 2 The cone of a normal affine toric variety - 2.1 Construction of the cone - 2.2 Proof of Theorem 1 - 3 Intersection Theory - 3.1 Prüfer v-multiplication rings - 3.2 Intersection theory with divisors - 3.3 T-invariant neighborhoods and Cartier divisors - 4 T-Linearization - 4.1 T-Linearization of a line bundle - 4.2 Existence of the T-linearization - 4.3 Ampleness - 5 Proof of Sumihiro’s theorem - 5.1 Construction of the T-invariant neighbourhood - 5.2 Classification - A Convex Geometrynrpages: 64status: publishe