Classification of Rational Unicuspidal Projective Curves Whose Singularities Have One Puiseux Pair

It is a very old and interesting open problem to characterize those collections of embedded topological types of local plane curve singularities which may appear as singularities of a projective plane curve C of degree d. (We invite the reader to consult the articles of Fenske, Flenner, Orevkov, Tono, Zaidenberg, Yoshihara, and the references therein, for recent developments.) The goal of the present article is to give a complete (topological) classification of those cases when C is rational and it has a unique singularity which is locally irreducible (i.e. C is unicuspidal) with one Puiseux pair. In fact, as a second goal, we also wish to present some of the techniques which are/might be helpful in such a classification, and we invite the reader to join us in our effort to produce a classification for all the cuspidal rational plane curves. In fact, this effort also motivates that decision, that in some cases (in order to have a better understanding of the present situation), we produce more different arguments for some of the steps. In the next paragraph we formulate the main result. We will write d for the degree of C and (a, b) for the Puiseux pair of its cusp, where 1 < a < b. We denote by {φj}j≥0 the Fibonacci numbers φ0 = 0, φ1 = 1, φj+2 = φj+1 + φj .