Geometry of hyperfields

TL;DR

This paper explores the topology of rational points over hyperfields, revealing deep connections between hyperfield structures and classical geometric spaces like schemes, Berkovich analytification, and real schemes.

Contribution

It establishes natural topologies on $H$-rational points for various hyperfields and proves homeomorphisms to classical geometric spaces, linking hyperfield geometry to established frameworks.

Findings

$X(H)$ is homeomorphic to the underlying space of $X$ for the Krasner hyperfield.

$X(H)$ is homeomorphic to the Berkovich analytification $X^{ extrm{an}}$ for the tropical hyperfield.

$X(H)$ is homeomorphic to the real scheme $X_r$ for the hyperfield of signs.

Abstract

Given a scheme $X$ over $\mathbb{Z}$ and a hyperfield $H$ which is equipped with topology, we endow the set $X(H)$ of $H$-rational points with a natural topology. We then prove that; (1) when $H$ is the Krasner hyperfield, $X(H)$ is homeomorphic to the underlying space of $X$, (2) when $H$ is the tropical hyperfield and $X$ is of finite type over a complete non-Archimedean valued field $k$, $X(H)$ is homeomorphic to the underlying space of the Berkovich analytificaiton $X^{\textrm{an}}$ of $X$, and (3) when $H$ is the hyperfield of signs, $X(H)$ is homeomorphic to the underlying space of the real scheme $X_r$ associated with $X$.