Places of algebraic function fields in arbitrary characteristic

TL;DR

This paper explores the structure of the space of all places of an algebraic function field in any characteristic, revealing density properties, characterizations of large fields, and conditions for existential closure, with topological insights.

Contribution

It provides new characterizations of large fields, analyzes the density of special places in the Zariski space, and investigates existential closure related to rational places.

Findings

Certain sets of places are dense in the Zariski topology.

Equivalent conditions characterize large fields.

Existential closure relates to the existence of rational places.

Abstract

We consider the Zariski space of all places of an algebraic function field $F|K$ of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zero-dimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of fields that are large, in the sense of F. Pop's Annals paper {\it Embedding problems over large fields}. We also study the question whether a field $K$ is existentially closed in an extension field $L$ if $L$ admits a $K$-rational place. In the appendix, we prove the fact that the Zariski space with the Zariski topology is quasi-compact and that it is a spectral space.