26 jun 2018 -- 11:30
Aula Tricerri, DiMaI, Firenze
If k is a finitely generated field extension of the rationals, and X is a smooth, proper, geometrically connected curve over k, any k-rational point of X provides a splitting of the canonical map from the algebraic fundamental group of X to the absolute Galois group Gal(k) of k. The so-called Grothendieck section conjecture essentially states the if X has genus at least 2 (i.e. it is hyperbolic), then such splittings are in bijection with k-rational points of X. Vistoli observed that, if Grothendieck's section conjecture is true, there should be some notion of dimension such that for an hyperbolic curve over k, the space of sections of the étale homotopy exact sequence has dimension 1. If we can prove this, we have a dimensional obstruction to the existence of sections. We propose such a notion as a modification of essential dimension, and make some steps toward proving that this dimension is 1 for the space of sections of P1 minus three points.