Abstract: Let X be a smooth projective curve over the complex numbers. The topological space of complex points of X is fairly simple: It is a one-point union of spheres, at least up to stable homotopy equivalence. If X is a smooth projective curve over an arbitrary field, one may consider it within the motivic homotopy theory of Morel and Voevodsky. Under the assumption that X has a rational point, it is possible to split off a top-dimensional sphere if and only if the tangent bundle of X admits a square root.