transactions of the american mathematical society Volume 342, Number 2, April 1994 AN ALMOST STRONGLYMINIMAL NON-DESARGUESIAN PROJECTIVE PLANE JOHN T. BALDWIN Abstract. There is an almost strongly minimal projective plane which is not Desarguesian. Zil'ber conjectured that every strongly minimal set is 'trivial', 'field-like', or 'module-like'. This conjecture was refuted by Hrushovski [4]. Varying his con- struction, we refute here two more precise versions of the conjecture. Zil'ber [8] calls a strongly minimal set M field-like if there is a pseudoplane definable in M. (A pseudoplane is an incidence structure such that each pair of lines intersect in only finitely many points and dually there are only finitely many lines passing through a pair of points.) This nomenclature would have been justified if the following conjecture were correct. Conjecture B of [8]. Every uncountably categorical pseudoplane is definable in an algebraically closed field and the field is definable in the pseudoplane. This conjecture has several aspects. In a model theoretic vein it limits the variety of uncountably categorical pseudoplanes. In particular, it asserts that there are only countably many such (as each must be defined in an algebraically closed field). This aspect of the conjecture is already refuted by [4]. (Hrushovski shows there are 2N° strongly minimal sets which are not locally modular. By Zil'ber's trichotomy theorem they are thus field-like.) A more geometric aspect of the problems is phrased in another Zil'ber con- jecture. Conjecture C of [8]. Every uncountably categorical affine plane is Desarguesian and hence is an affine plane over an algebrically closed field. We show that the projective plane constructed here does not interpret a group and thus cannot be Desarguesian. The affine plane associated with this projec- tive plane also fails to be Desarguesian so Conjecture C is refuted. Finally there is a more algebraic geometric conjecture behind the part of Conjecture B asserting every field-like structure is definable in a field. Received by the editors October 23, 1989 and, in revised form, May 14, 1992. 1991 Mathematics Subject Classification.Primary 03C60; Secondary 51C10. Partially supported by NSF grant 8602558. In the time since this paper was submitted (October 1989) various authors, including Herwig, Poizat, and Wagner, have suggested alternative methods to organize the proof that the generic model is (o-saturated and that no infinite group is interpretable in a structure constructed by Hrushovski's method. © 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page 695 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use