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A projective invariant for swallowtails and godrons, and global theorems on the flecnodal curve. (English) Zbl 1120.58026
Summary: 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).

##### MSC:
 58K35 Catastrophe theory 58K60 Deformation of singularities 14B05 Singularities in algebraic geometry 32S25 Complex surface and hypersurface singularities 53A20 Projective differential geometry 53A15 Affine differential geometry 53A05 Surfaces in Euclidean and related spaces 53D99 Symplectic geometry, contact geometry 70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
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