[Paper Review] On relative birational geometry and Nagata's compactification

Michael Temkin, Ilya Tyomkin|arXiv (Cornell University)|Nov 25, 2013

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper extends relative birational geometry to algebraic spaces using a model-theoretic framework, introducing semivaluation spaces and pushouts to provide a new proof of Nagata’s compactification theorem for algebraic spaces. It establishes a valuative criterion for schematization and characterizes Prüfer morphisms, offering foundational tools for generalizations to stacks and equivariant geometry.

ABSTRACT

In 2011, the first author introduced (relative) Riemann-Zariski spaces corresponding to a morphism of schemes and established their basic properties. In this paper we clarify that theory and extend it to morphisms between algebraic spaces. As an application, a new proof of Nagata's compactification theorem for algebraic spaces is obtained.

Motivation & Objective

• To extend the theory of relative Riemann-Zariski spaces from schemes to algebraic spaces.
• To provide a new, model-theoretic proof of Nagata’s compactification theorem for algebraic spaces.
• To develop a valuative criterion for schematization of algebraic spaces using semivaluation spaces.
• To characterize Prüfer morphisms and pairs in the context of qcqs algebraic spaces.
• To lay the groundwork for future generalizations to equivariant geometry and Deligne-Mumford stacks.

Proposed method

• Introduces the concept of models as triples $\mathbf{X} = (\mathcal{X}, X, \psi_\mathbf{X})$ where $\psi_\mathbf{X}: \mathcal{X} \to X$ is a separated schematically dominant morphism.
• Defines modifications of models as morphisms where the $\mathcal{X}$-map is an isomorphism and the $X$-map is proper.
• Uses semivaluation spaces and their pushouts to analyze the valuative structure of models.
• Applies Ferrand pushouts and approximation techniques to construct schemes from algebraic spaces via blow-ups.
• Employs quasi-modifications and quasi-blow-ups to reduce global schematization problems to local valuative conditions.
• Leverages the quasi-compactness of the valuative space $\mathrm{Val}(\mathbf{X})$ to cover the space with finitely many scheme-like pieces.

Experimental results

Research questions

• RQ1 Can the theory of relative Riemann-Zariski spaces be extended from schemes to algebraic spaces?
• RQ2 How can Nagata’s compactification theorem be re-proven using a model-theoretic and valuative approach?
• RQ3 What valuative conditions ensure that an algebraic space becomes a scheme after an $\mathcal{X}$-admissible blow-up?
• RQ4 What characterizes Prüfer morphisms between qcqs algebraic spaces in terms of modifications and schematization?
• RQ5 How do semivaluation spaces and pushouts facilitate the construction of compactifications and schematizations?

Key findings

• A new proof of Nagata’s compactification theorem for algebraic spaces is obtained via the existence of affine modifications for any model.
• Any separated morphism of algebraic spaces factors as a composition of an affine morphism followed by a proper morphism, generalizing the classical factorization result.
• An algebraic space $X$ with a schematically dense quasi-compact open subscheme $\mathcal{X}$ admits an $\mathcal{X}$-admissible blow-up to a scheme if and only if all valuation spaces $T$ associated to adic semivaluations $\mathbf{T} \to \mathbf{X}$ are separated.
• A model $\mathbf{X}$ is Prüfer if and only if $\psi_\mathbf{X}$ is a pro-open immersion and $(X, \mathcal{X})$ is a Prüfer pair.
• The class of Prüfer morphisms between qcqs algebraic spaces is stable under étale base changes.
• The valuative space $\mathrm{Val}(\mathbf{X})$ is quasi-compact, enabling finite covers by scheme-like pieces via quasi-blow-ups.