2-Nilpotent real section conjecture
We show a 2-nilpotent section conjecture over ℝ: for a geometrically connected curve X over ℝ such that each irreducible component of its normalization has ℝ-points, π0(X(ℝ)) is determined by the maximal 2-nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that for X smooth and proper, X(ℝ)± is determined by themaximal 2-nilpotent quotient of Gal(ℂ(X)) with its Gal(ℝ) action, where X(ℝ)± denotes the set of real points equipped with a real tangent direction, showing a 2-nilpotent birational real section conjecture. © 2013 Springer-Verlag Berlin Heidelberg.
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