The geometry of weakly minimal types

Abstract

AbstractLet T be superstable. We say a type p is weakly minimal if R(p, L, ∞) = 1. Let M ⊨ T be uncountable and saturated, H = p(M). We say D ⊂ H is locally modular if for all X, Y ⊂ D with X = acl(X) ∩ D, Y = acl(Y) ∩ D and X ∩ Y ≠ ∅,Theorem 1. Let p ∈ S(A) be weakly minimal and D the realizations of stp(a/A) for some a realizing p. Then D is locally modular or p has Morley rank 1.Theorem 2. Let H, G be definable over some finite A, weakly minimal, locally modular and nonorthogonal. Then for all a ∈ H∖acl(A), b ∈ G∖acl(A) there area′ ∈ H, b′ ∈ G such that a′ ∈ acl(abb′A)∖acl(aA). Similarly when H and G are the realizations of complete types or strong types over A.