  We exhibit sufficient conditions for a finite collection of periodic orbits
of a Reeb flow on a closed $3$-manifold to bound a positive global surface of
section with genus zero. These conditions turn out to be $C^\infty$-generically
necessary. Moreover, they involve linking assumptions on periodic orbits with
Conley-Zehnder index ranging in a finite set determined by the ambient contact
geometry. As an application, we reprove and generalize a classical result of
Birkhoff on the existence of annulus-like global surfaces of section for
geodesic flows on positively curved two-spheres.
