The geometry of 1-based minimal types

In this paper, we study the geometry of a (nontrivial) 1-based 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 -categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.