An almost strongly minimal non-Desarguesian projective plane

John T. Baldwin
1994 Transactions of the American Mathematical Society  
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 construction, 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
more » ... tely 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. 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.
doi:10.1090/s0002-9947-1994-1165085-8 fatcat:qsx7wdkzajchhavyq4wnhrnhtm