Title: Two results about Grothendieck's Section Conjecture, Speaker(s): Giulio Bresciani, SNS Pisa, Date and time: 18 May 2022, 14:30 (Europe/Rome), Lecture series: Algebraic and Arithmetic Geometry Seminar, Venue: Dipartimento di Matematica (Aula Magna).
You can access the full event here: https://events.dm.unipi.it/e/73
Abstract --------
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 $X$.
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