Merging divisorial with colored fans

Gluing Affine Torus Actions via Divisorial Fans

Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a “proper polyhedral divisor” introduced in earlier work, we develop the concept of a “divisorial fan” and show that these objects encode the equivariant gluing of affine varieties with torus action. We ...

Divisorial rings and Cox rings

1 Preliminaries on monoids Definition 1.1. An Abelian monoid is a set with a binary, associative, and commutative operation which has a neutral element. It will often be called just a monoid in this manuscript because we will not deal with non-commutative monoids. A monoid M is called • finitely generated if there is a finite set of generators, or equivalently if there is a surjection of monoid...

Quotients of Divisorial Toric

A divisorial valuation with irrational volume

of an m-primary ideal a⊆ R. In fact, if qm = a for a fixed ideal a then it is evident that e(a)= vol(v). The volume of a valuation has implicitly been studied already in [4], but it was first explicitly defined in [5]. The terminology is intended to emphasize the relation with global invariants of linear series on projective varieties. A natural question is to what extent the properties of vol(...

Quotients of Divisorial Toric Varieties

We consider subtorus actions on divisorial toric varieties. Here divisoriality means that the variety has many Cartier divisors like quasiprojective and smooth ones. We characterize when a subtorus action on such a toric variety admits a categorical quotient in the category of divisorial varieties. Our result generalizes previous statements for the quasiprojective case. A first step in the proo...