Cox rings of morphisms and resolution of singularities

TL;DR

This paper extends Cox ring constructions to proper birational morphisms of schemes, enabling new methods for resolving singularities with applications across different characteristics.

Contribution

It generalizes Cox ring constructions to a broader class of morphisms and introduces new tools for resolving singularities in algebraic geometry.

Findings

Extended Cox ring construction to proper birational morphisms

Representation of morphisms via schemes with torus actions

Applications to resolution of singularities in arbitrary characteristic

Abstract

We extend the Cox-Hu-Keel construction of the Cox rings to any proper birational morphisms of normal noetherian schemes. It allows the representation of any proper birational morphism by a map of schemes with mild singularities with torus actions. In a particular case, the notion generalizes the combinatorial construction of Satriano and the recent construction of multiple weighted blow-ups on Artin-stacks by Abramovich-Quek. The latter can be viewed as an extension of stack theoretic blow-ups by Abramovich, Temkin, and Wlodarczyk, a similar construction of McQuillan and the author's recent cobordant blow-ups at weighted centers to a more general situation of arbitrary locally monomial centers. We show some applications of this operation to the resolution of singularities over a field of any characteristic.