A Zariski geometry is a strongly minimal structure Z for which the class of positive quantifier-free (parametrically-) definable sets gives Z and its Cartesian powers a "nice" sequences of topologies. More precisely, we require that
The main theorem on Zariski geometries, due to Hrushovski and Zilber, is that a non-locally modular Zariski geometry interprets an algebraically closed field, and does so, at least morally, in terms of tangency. However, it may happen that a Zariski geometry is not interpretable in any field.
The class of Zariski-type structures generalizes that of Zariski geometries by allowing for structures of higher dimension and (in some cases) relaxing the condition on dimensions of intersections. While in the case of Zariski geometries there is only one reasonable choice for the dimension function (because Morley rank, algebraic dimension, Noetherian dimension, et cetera all agree), in axiomatizing general Zariski-type structures, it is necessary to include a dimension function as part of the data.