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Vol. 15 No. 1 (2024)
Vol 15, Núm.1 (2024): Enero-Junio

Revisiones/Reviews

A review of one-dimensional Zariski geometries

https://doi.org/10.19053/01217488.v15.n1.2024.16237

Published 2024-02-12

  • Joel Torres del Valle
    • ,
  • Pedro Rizzo

Joel Torres del Valle
a:1:{s:5:"es_ES";s:24:"Universidad de Antioquia";}

Pedro Rizzo
Universidad de Antioquia

Abstract

In this article, we present a comprehensive review of Zariski geometries of dimension one.We begin with a brief retrospective of key concepts in model theory and its historical evolution. Subsequently, we introduce concepts in algebraic geometry, exploring their intersection with model theory to establish geometric model theory. We emphasize Zilber’s Trichotomy Conjecture and how Hrushovski’s refutation led Zilber and Hrushovski to isolate the class of structures where the conjecture holds, giving rise to Zariski geometries. We conclude by presenting the results obtained by Zilber and Hrushovski in the 90s and provide an updated bibliographic review of the topic.

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