“Projective curves and weak second-order logic”
Venerdì 30 Maggio 2025, ore 14:30 - Sala Riunioni 7B1 - Francesco Gallinaro (Università di Pisa)
Abstract
Let K be an algebraically closed field. Answering a question of Tressl, we prove that the incidence relation between points and irreducible curves in the projective plane over K is bi-interpretable with the weak second-order theory of K, and it is therefore undecidable. We show that this theory depends on the transcendence degree of K, so that for example the incidence relation over the complex field and the incidence relation over the field of algebraic numbers have different theories.
This leads us to a study of the weak second-order theory of the complex field: we show that every definable subset of the embedded copy of the integers is definable in the pure ring structure on Z, in stark contrast with the weak second-order theory of the reals, and we give an axiomatization which is recursive modulo the theory of the embedded copy of the integers.
This is joint work with Alessandro Berarducci.
