On the Theory of Specialisations of Regular Covers of Zariski Structures

Publication date: 16 February 2023

Abstract: In algebraic geometry specialisations and valuations play and important role. In this paper we start investigating analogous structures for Zariski structures. Specifically, we look into the existence and uniqueness properties of extensions of universal specialisations from a base Zariski structure to its regular cover. In the process we begin to uncover some structural properties of regular covers of Zariski structures, and also to uncover the type of topological properties necessary for a Zariski structure to have a "good" theory of specialisations. A subclass of Zariski structures is identified with a ``good theory of specialisations.

Mathematics Subject Classification ID

Topology of vector bundles and fiber bundles (57R22) Generalizations of fiber spaces and bundles in algebraic topology (55R65) Models with special properties (saturated, rigid, etc.) (03C50) Properties of classes of models (03C52) Basic properties of first-order languages and structures (03C07) Quantifier elimination, model completeness, and related topics (03C10) Categoricity and completeness of theories (03C35) Foundations of algebraic geometry (14A99) Abstract manifolds and fiber bundles (category-theoretic aspects) (18F15)

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