On the Number of the Cusps of Rational Cuspidal Plane Curves

A cuspidal curve is a curve whose singularities are all cusps, i.e. unibranched singularities. The article describes computations which lead to the following conjecture: A rational cuspidal plane curve of degree greater or equal to six has at most three cusps. The curves with precisely three cusps occur in three series. Assuming the Flenner–Zaidenberg rigidity conjecture the above conjecture is verified up to degree 20. Let C ⊂ P be a plane curve. The curve is called cuspidal if all its singularities are cusps, i.e., unibranched singularities. It is known that rational cuspidal plane curves underlie severe restrictions. For example Tono proved [T]: Theorem 1 A rational cuspidal plane curve has at most eight cusps. It is not known whether this bound is sharp. In fact, one of the main purposes of this article is to provide evidence for the following conjecture: Conjecture 2 A rational cuspidal plane curve has at most three cusps — with the exception of a rational plane quintic with four cusps. We will describe the topological type of a cusp singularity by its multiplicity sequence, m = (m0,m1, . . . ,mn): Let V := Xn πn −→ Xn−1 −→ · · · −→ X1 π1 −→ X0 = (C, 0) ∪ ∪ ∪ ∪ Cn −→ Cn−1 −→ · · · −→ C1 −→ C0 = (C, p) be the minimal embedded resolution of the curve germ (C, p) such that the total inverse image D := π−1 n ◦ · · · ◦ π−1 1 (C) of the curve is a normal crossing divisor. Then the multiplicity sequence consists of the multiplicities mi of the strict transforms Ci of the curve germs. For example a simple cusp has the multiplicity sequence (2, 1, 1). To shorten the notation, sequences of the same multiplicity in the multiplicity sequence are indicated by indices, e.g., (7, 23, 12) means (7, 2, 2, 2, 1, 1). (Here we follow the notation of [dJP] and [MS89], i.e., the number of 1s at the end of the multiplicity sequence equals the smallest number greater than one in the multiplicity sequence. [FZ96, FZ00] add an additional