#### Title

The Zariski-Riemann Space as a Universal Model for the Birational Geometry of a Function Field

#### Date of Degree

6-2022

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Program

Mathematics

#### Advisor

Hans Schoutens

#### Committee Members

Laura Ghezzi

Andrew Obus

#### Subject Categories

Algebra | Algebraic Geometry

#### Keywords

algebra, algebraic geometry, valuation theory, scheme theory, Zariski-Riemann space

#### Abstract

Given a function field $K$ over an algebraically closed field $k$, we propose to use the Zariski-Riemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between ad-hoc collections of open subsets of $\ZR (K/k)$ ordered by quasi-refinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal model. Conversely, given a sheaf on a normal model, we show how to extend it to a sheaf on $\ZR (K/k)$ and we prove that both operations are compatible with pushforwards and pullbacks. As a result, we study the cohomology theory of the Zariski-Riemann space and we show that, on each local model, it yields the usual sheaf cohomology. Lastly, we focus on ideal-sheaves and we interpret blow-ups using our framework.

#### Recommended Citation

Pignatti Morano di Custoza, Giovan Battista, "The Zariski-Riemann Space as a Universal Model for the Birational Geometry of a Function Field" (2022). *CUNY Academic Works.*

https://academicworks.cuny.edu/gc_etds/4775