2-Nilpotent Real Section Conjecture

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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, pi₀(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 the maximal 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.