Abstract
AbstractLet $X$
X
be a curve over a field $k$
k
finitely generated over ℚ and $t$
t
an indeterminate. We prove that, if $s$
s
is a section of $\pi _{1}(X)\to \operatorname{Gal}(k)$
π
1
(
X
)
→
Gal
(
k
)
such that the base change $s_{k(t)}$
s
k
(
t
)
is birationally liftable, then $s$
s
comes from geometry. As a consequence we prove that the section conjecture is equivalent to the cuspidalization of all sections over all finitely generated fields.
Publisher
Springer Science and Business Media LLC
Reference23 articles.
1. Abramovich, D., Graber, T., Vistoli, A.: Gromov-Witten theory of Deligne-Mumford stacks. Am. J. Math. 130(5), 1337–1398 (2008)
2. Anderson, M.P.: Exactness properties of profinite completion functors. Topology 13(3), 229–239 (1974)
3. Borne, N., Vistoli, A.: The Nori fundamental gerbe of a fibered category. J. Algebraic Geom. 24, 311–353 (2015)
4. Borne, N., Vistoli, A.: Fundamental gerbes. Algebra Number Theory 13(3), 531–576 (2019)
5. Bresciani, G.: Essential dimension and pro-finite group schemes. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 22(4), 1899–1936 (2021)