Intersection norms on surfaces and Birkhoff sections for geodesic flows

Every filling multicurve on a smooth surface determines a norm on the first homology group of the surface.The unit ball of the dual norm is the convex hull of finitely many integer points.We give an interpretation of these points in terms of certain coorientations of the multicurve.Our main result is a classification statement: when the surface is hyperbolic and the filling multicurve is geodesic, integer points in the interior of the unit ball of the dual norm classify isotopy classes of Birkhoff sections for the geodesic flow (on the unit tangent bundle to the surface) whose boundary is the symmetric lift of the multicurve.All results remain true when one replaces the hyperbolic surface by a 2-dimensional orientable hyperbolic orbifold.