Projective curves and weak second-order logic

Given an algebraically closed field [Formula: see text] of characteristic zero, we study the incidence relation between points and irreducible projective curves, or more precisely the poset of irreducible proper subvarieties of [Formula: see text]. Answering a question of Marcus Tressl, we prove that the poset interprets the field, and it is in fact bi-interpretable with the two-sorted structure consisting of the field [Formula: see text] and a sort for its finite subsets. In this structure, one can define the integers, so the theory is undecidable. When [Formula: see text] is the field of complex numbers we can nevertheless obtain a recursive axiomatization modulo the theory of the integers. We also show that the integers are stably embedded and that the poset of irreducible varieties over the complex numbers is not elementarily equivalent to the one over the algebraic numbers.