Dipartimento di Matematica, Aula Magna.<\/p>\n
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 $X$.<\/p>\n
Further information is available on the event page<\/a> on the Indico platform.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 $X$.…<\/p>\n