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## Geometries of Hrushovski constructions

In 1984 Zilber conjectured that any strongly minimal structure is geometrically equivalent to one of the following types of strongly minimal structures, in the appropriate language: Pure sets, Vector Spaces over a fixed Division Ring and Algebraically Closed Fields. In 1993, in his article `A new strongly minimal set' Hrushovski produced a family of counterexamples to Zilber's conjecture. His method consists in two steps. Firstly he builds a `limit' structure from a suitable class of finite structures in a language consisting only of a ternary relational symbol. Secondly, in a step called the collapse, he defines a continuum of subclasses such that the corresponding `limit' structures are new strongly minimal structures. These new strongly minimal structures are non isomorphic but Hrushovski then asks if they are geometrically equivalent. We first analyze the pregeometries arising from different variations of the construction before the collapse. In particular we prove that if we repeat the construction starting with an n-ary relational symbol instead of a 3-ary relational symbol, then the pregeometries associated to the corresponding `limit' structures are not locally isomorphic when we vary the arity. Second we prove that these new strongly minimal structures are geometrically equivalent. In fact we prove that their geometries are isomorphic to the geometry of the `limit' structure obtained before the collapse.

Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:514393 |

Date | January 2009 |

Creators | Ferreira, Marco Antonio Semana |

Contributors | Evans, David |

Publisher | University of East Anglia |

Source Sets | Ethos UK |

Detected Language | English |

Type | Electronic Thesis or Dissertation |

Source | https://ueaeprints.uea.ac.uk/20002/ |

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