If the assumption that the geometry is locally finite is dropped, the author shows that every strongly minimal set falls into one of the above categories or contains a definable pseudoplane, a combinatorial configuration introduced by Lachlan in a different model-theoretic context. (Usually, classes (ii) and (iii) are combined in this context to form the class of module-like structures, so this is called the trichotomy theorem.) The author conjectured that those strongly minimal sets which interpret pseudoplanes are, roughly speaking, bi-interpretable with an algebraically closed field. This conjecture was refuted by Hrushovski. Hrushovski and the author have developed the class of Zariski structures---geometries with sufficient structure to recover the conjecture. This development displays intensive connections between model theory and algebraic geometry.
(http://www.cs.bgu.ac.il/~kojman/colloquium/zilberrev.html)