A projective invariant for swallowtails and godrons, and global theorems on the flecnodal curve

We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a godron (called also cusp of Gauss): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these properties and a projective invariant that we associate to each godron we present all possible local configurations of the tangent plane, the self-intersection line, the cuspidal edge and the flecnodal curve at a generic swallowtail in $\mathbb R^3$. We present some global results, for instance: In a hyperbolic disc of a generic smooth surface, the flecnodal curve has an odd number of transverse self-intersections (hence at least one self-intersection).