I am currently reading David Marker's paper "A Remark on Zilber's Pseudoexponentiation" (J. Symb. Logic 71 (2006), no. 3). He describes the axioms for Zilber's pseudoexponential fields. The Strong Exponential Closure axiom is stated as follows:
For all finite $A$ if $V\subseteq G_n(K)$ is irreducible, free and normal there is $(\bar{x},E(\bar{x}))\in V$ a generic point of $V$ over $A$.
$A$ is any finite subset of $K$.
$G_n(K)=K\times K^{\times}$ where $K$ is the field, and $E$ is the exponential function on the field.
My question is what it means that a point is generic over a set. It seems to be an expression from algebraic geometry, which I don't know much about. I have already found out what a generic point is (using the Zariski Topology), but cannot find anywhere what it means for a point to be generic over a set.
The overview of Zilber fields given in this paper adequately explains what the "generic point" language in that axiom means. The term is used in the sense of algebraic geometry but it requires quite a few preliminary definitions to unpack.