### Abstract

In this paper, we study the geometry of a (nontrivial) 1-based SU rank-1 complete type. We show that if the (localized, resp.) geometry of the type is modular, then the (localized, resp.) geometry is projective over a division ring. However, unlike the stable case, we construct a locally modular type that is not affine. For the general 1-based case, we prove that even if the geometry of the type itself is not projective over a division ring, it is when we consider a 2-fold or 3-fold of the geometry altogether. In particular, it follows that in any ω-categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.

Original language | English |
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Pages (from-to) | 4241-4263 |

Number of pages | 23 |

Journal | Transactions of the American Mathematical Society |

Volume | 355 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2003 Oct |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

De Piro, T., & Kim, B. (2003). The geometry of 1-based minimal types.

*Transactions of the American Mathematical Society*,*355*(10), 4241-4263. https://doi.org/10.1090/S0002-9947-03-03327-0