The geometry of Hrushovski constructions, I. The uncollapsed case

TL;DR

This paper investigates the geometric properties of certain omega-stable structures arising from Hrushovski's construction, focusing on how language variations influence their pregeometries.

Contribution

It analyzes the pregeometries associated with the regular type of rank omega in Hrushovski constructions, revealing invariance in finite subpregeometries despite language changes.

Findings

Varying the language L affects the local isomorphism type of the pregeometry.

Finite subpregeometries remain unchanged across different languages.

The study provides insights into the geometric structure of intermediate Hrushovski constructions.

Abstract

An intermediate stage in Hrushovski's construction of flat strongly minimal structures in a relational language L produces omega-stable structures of rank omega. We analyze the pregeometries given by forking on the regular type of rank omega in these structures. We show that varying L can affect the (local) isomorphism type of the pregeometry, but not its finite subpregeometries. A sequel will compare these to the pregeometries of the strongly minimal structures.