Fonctionne avec Indico
v3.2.9
In an attempt to classify the geometry of strongly minimal sets, Zilber had conjectured them to split into three different types: Trivial geometries, geometries which are vector space like and those which are field like. Hrushovski later refuted this conjecture by introducing a clever construction that had been modified and used a lot ever since. His counterexample to Zilbers conjecture provided a structure, which was not one-based, so could not be of trivial or vector space type, but nevertheless it forbade a certain point-line-plane configuration, which is present in fields. Hrushovski called that property CM-triviality and later Pillay, with some corrections by Evans, defined a whole hierarchy of new geometries, on which's base we find non-one basedness (1 ample) and non-CM-triviality (2-ample) and on which's top we find fields, being n-ample for all n. Recently, Baudisch, Pizarro and Ziegler and independently Tent have provided examples proving that this ample hierarchy is strict. While their examples are omega-stable of infinite rank, it remained open if one can find geometries of finite rank which are at least 2-ample but do not interpret a field. In this talk we will now introduce an almost strongly minimal structure which is 2-ample, but not 3-ample, using a Hrushovski-like construction. This is a joint work with K. Tent.