Throughout this chapter, we assume unless stated otherwise that our Zariski geometry is C, the one-dimensional, irreducible, pre-smooth Zariski structure satisfying (EU) which we studied in Section 3.8. We follow the notation and assumptions of that sub-section. Our main goal is to classify such structures, which is essentially achieved in Theorem 4.4.1. In fact, the proof of the main theorem deepens the analogy between our abstract Zariski geometry and algebraic geometry. In particular, we prove generalisations of Chaos theorem on analytic subsets of projective varieties and of Bezout's theorem. As a by-product, we develop the theory of groups and fields living in pre-smooth Zariski structures.

Getting a group

Our aim in this section is to obtain a Zariski group structure living in C. The main steps of this construction are as follows:

• We consider the composition of local functions Va → Va (branches of curves through 〈a, a〉) modulo the tangency and show that generically it defines an associative operation on a pre-smooth Zariski set, a pre-group of jets.

• We prove that any Zariski pre-group can be extended to a group with a pre-smooth Zariski structure on it. This is an analogue of Weil's theorem on group chunks in algebraic geometry.

We consider tangency of branches of curves through 〈b, a〉, 〈a, a〉, and potentially through other points on C2. We keep the notation T for this tangency as well, when there is no ambiguity about the point at which the branches are considered.