Some Questions concerning Hrushovski’s Amalgamation Constructions

In his book on stable groups [Poi87] Poizat writes (with respect to our 1987 understanding of ω-stable fields): ”Nous n’avons pas fait de progrès décisif depuis le temps oú nous chassions les aurochs avec une hache de pierre; nous avons seulment acquis une meilleure comprehension de l’enjeu du problem”. Sadly, there is no better way to describe the current situation with respect to the problem of classifying strongly minimal sets. Since the refutation of Zilber’s Conjecture, Hrushovski’s amalgamation construction of new strongly minimal sets, introduced in [Hru93] and [Hru92], stood in the way of any (naive) attempt of classification of their possible pregeometries. Indeed, our mastery of the techniques underlying these constructions is now better than it used to be, but our understanding of the fundamental questions they give rise to can hardly be said to have improved. To the best of my knowledge, there has not been any progress at all with respect to some questions that have already been asked in Hrushovski’s original paper. Several survey papers deal with these amalgamation constructions (e.g. [Wag94], [Bal02] and [Poi02]). These papers are concerned mainly with the construction itself developing axiomatic frameworks in which it can be carried out and the known examples it gives rise to. The aim of the present paper is threefold, and different: • Present a setting given in the language of (standard) geometric stability theory in which these construction can be understood. • Show how new structures which have been recently constructed using these methods (most notably [BMPZ05b] and [BHMPW06]) fit into this framework. • Point out fundamental questions concerning these constructions, that to the best of my knowledge have not been addressed. A central theme of this paper is that the major gap in our understanding of the scope of these constructions lies in the technically simpler part, usually known as the free construction, whereas the more involved part of the construction, known as the collapse, is fairly well understood.