The Schubert Triangle Geometry on an Algebraic Surface

For a smooth complex projective surface, and for two families of curves with traditional singularities in it, we enumerate the pairs of curves in each family having two points of contact among them, thus generalizing the double contact formulae known or conjectured by Zeuthen and Schubert in the case of the complex projective plane. The technique we use to this purpose is a particular notion of triangle which can be defined in any smooth surface, thus potentially generalizing to arbitrary surfaces the Schubert technique of triangles.