Abstract

TORRES DEL VALLE, Joel  and  HERNANDEZ RIZZO, Pedro. A review of one-dimensional Zariski geometries. Ciencia en Desarrollo [online]. 2024, vol.15, n.1, pp.155-170.  Epub Oct 15, 2024. ISSN 0121-7488.  https://doi.org/10.19053/01217488.v15.n1.2024.16237.

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 Hmshovski's refutation led Zilber and Hmshovski 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 Hmshovski in the 90s and provide an updated bibliographic review of the topic.

Keywords : Zariski geometry; model theory; algebraic geometry.

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