LetR=⊕g∈GRgbe aG-graded ring. In this paper we define the homogeneousequivalence concept between graded rings. We discuss some properties of theG-graded rings and investigate which of these are preserved under homogeneous-equivalence maps. Furthermore, we give some results in graded ring theory and also some applications of this concept toZ-graded rings.
We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and delivers all Mori dream spaces with torus action. We exhibit the example class of ‘general arrangement varieties’ and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.
Categorical methods in graded ring theory
