Relevant maps and the algebraic skeleton of simplicial toric prevarieties
Felix Goebler
TL;DR
The paper develops a comprehensive framework for rational maps between multigraded Proj schemes by introducing and leveraging conical rings and relevant subsets. It shows that a robust notion of maps between multigraded rings (including relevant, rational, and conical data) yields a precise translation between ring-theoretic data and geometric morphisms, culminating in an anti-equivalence between the category of simplicial toric prevarieties and a category of conical rings with rational maps. It further demonstrates that multigraded noetherian polynomial rings encode systems of fans, enabling a reverse Cox-type correspondence that recovers toric prevarieties from ring data and clarifies how to refine Cox rings via subrings. The results unify Proj constructions and Cox ring methods, extend to subtorus actions, and show that Chow quotients of simplicial toric prevarieties remain simplicial toric prevarieties, providing a solid algebraic backbone for toric geometry beyond normal projective varieties.
Abstract
Morphisms between schemes arising from multigraded rings are essential for understanding geometric relationships in algebraic geometry, yet a systematic theory for such maps has been lacking. In this paper, we develop a comprehensive framework for rational maps between multigraded Proj schemes by introducing several notions of maps between their underlying multigraded rings. A key challenge is that to induce actual morphisms (rather than just rational maps), the ring homomorphism $\varphi\colon R \to S$ must hit every relevant element in $S$. To address this, we introduce the use of relevant subsets $B \subseteq S_+$ (where $S_+$ is the ideal generated by all relevant elements), $B \unlhd S$, which allow us to control this condition more flexibly. As an application, we show that multigraded noetherian polynomial rings naturally encode combinatorial data, giving rise to systems of fans and thus to toric prevarieties. By leveraging our notion of rational maps with those relevant subsets, we prove that the category of triples $(D, S, B)$ - where $D$ is a finitely generated abelian group, $S$ is a $D$-graded noetherian polynomial ring, and $B \unlhd S$ is a subset of $S_+$ - together with rational maps of conical rings, is anti-equivalent to the category of simplicial toric prevarieties.