The modulus of unfoldings of cusps in conformal geometry

In this paper we give the complete classification of generic 1-parameter unfoldings of germs of real analytic curves with a cuspidal point under conformal equivalence. A cusp is obtained by squaring an analytic curve having contact of order 1 with a line through the origin. We show that this point of view can be extended to the unfolding. This allows to reduce the classification of unfoldings of cusps to the classification of unfoldings of a pair of curves having a contact of order 1 at the origin, one being obtained from the other through a reflection with respect to the origin. This unfolding can be studied in the same way as an unfolding of a curvilinear angle with zero angle, called a horn. We then classify the unfoldings of the special horns corresponding to cusps by means of the associated diffeomorphisms. We interpret the results geometrically.