| Résume | Joint work with Francesco Gallinaro. Addressing a question of Marcus Tressl, we study the structure Var(C) given by the incidence relation between points and irreducible complex projective curves. We show that Var(C) is bi-interpretable with the polynomial ring C[x] and therefore it has an undecidable theory, as it interprets both C and the subring Z of integers. Nevertheless Var(C) is tame modulo the integers, in the sense that a copy Z of the ring of integers is stably and conservatively embedded in Var(C), all types are controlled by the integers, and the theory of Var(C) is recursively axiomatizable over the complete theory of Z. By contrast, we do not know whether Var(R) is undecidable, or whether it interprets R, where R is the real field and Var(R) is defined like Var(C) but considering only the real points. Replacing C with the field K of complex algebraic numbers yields a structure Var(K) that is not elementary equivalent to Var(C) but it is again recursively axiomatizable over Z. Our results depend on a bi-interpretation of Var(C) with the two-sorted structure (C,Fin(C)) consisting of the field C and a sort for its finite subsets, with the membership relation between the two sorts. |
