Algebraic and Arithmetic Geometry Seminar

Two results about Grothendieck's Section Conjecture

by Giulio Bresciani (SNS Pisa)

Aula Magna (Dipartimento di Matematica)

Aula Magna

Dipartimento di Matematica


Grothendieck's Section Conjecture states that, if $X$ is an hyperbolic curve over a field $k$ finitely generated over $\mathbb{Q}$, every section of the map $\pi_1(X) \to \mathsf{Gal}(k)$ is associated with a rational point of the completion of $X$. After summarizing the main known facts concerning the conjecture, we will present two new results. First, we generalize to number fields a theorem which was proved over $\mathbb{Q}$ by Stix: we prove that if $k$ is a number field and the Weil restriction of $X$ to $\mathbb{Q}$ admits a rational map to a non-trivial Brauer-Severi variety, then $X$ satisfies the conjecture. Secondly, if $k$ is finitely generated over $\mathbb{Q}$, we prove that the conjecture holds for sections which satisfy a strong birationality assumption. In particular, this implies that the section conjecture is equivalent to Esnault and Hai's cuspidalization conjecture, which states that every Galois section of every hyperbolic curve $X$ lifts to every open subset of $X$.