Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s).

Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following:

(a) Trivial (No structure)

(b) Linear (Locally modular): Vector Spaces.

(c) Non-Linear (Non Locally modular): Algebraically Closed Fields(ACF).

In other words Zilber conjectured that any non-locally modular strongly minimal set interprets an ACF.

Hrushovski construct a new class of $\aleph_{1}$- categorical structures, disproving Zilber's conjecture.(A New Strongly Minimal Set - Annals of Pure and Applied Logic 1993 ).

Hrushovski generalized the Frasse's method to construct his counter example. Hrushovski construct an $\omega$- saturated strongly minimal set whose geometry is flat.

Definition: A combinatorial geometry J is flat if whenever $E_{i}(i\in I)$ are finite number of finite-dimensional closed subset of J, s ranges over the subsets of I, then $\sum_{s}(-1)^{card(s)}\leq0$.

Hrushovski proved the following theorem:

Theorem: Let D be a saturated strongly minimal set whose geometry is flat. Then D does not interpret an infinite group. So D does not interpret an ACF.

Question: Can we use Hrushovski's method to find a saturated strongly minimal set whose geometry is not flat ?

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