Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture

Abstract

Abstract Let E ⊆ ℙ 2 {E\subseteq\mathbb{P}^{2}} be a complex rational cuspidal curve and let ( X , D ) → ( ℙ 2 , E ) {(X,D)\to(\mathbb{P}^{2},E)} be the minimal log resolution of singularities. We prove that E has at most six cusps and we establish an effective version of the Zaidenberg finiteness conjecture (1994) concerning Eisenbud–Neumann diagrams of E. This is done by analyzing the Minimal Model Program run for the pair ( X , 1 2 ⁢ D ) {(X,\frac{1}{2}D)} . Namely, we show that ℙ 2 ∖ E {\mathbb{P}^{2}\setminus E} is ℂ * * {\mathbb{C}^{**}} -fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.