The section conjecture by A. Grothendieck proposes to look at rational points of curves through their decomposition subgroups in the arithmetic fundamental group of the curve, i.e., sections of the homotopy short exact sequence. And it furthermore predicts in the case of curves over number fields that any such section arises in this way from a rational point or a rational cusp. The talk will discuss that for curves over number fields or local fields the centralizer of a section is necessarily trivial and deduce two natural properties of the set of all sections, that would follow from the section conjecture when suitably generalized: (i) Galois descent for points, (ii) the space of all sections for a curve over local fields is profinite.