Cuspidal curves, minimal models and Zaidenberg's finiteness conjecture

Abstract

Let $E\subseteq {\mathbb{P}}^2$ be a complex rational cuspidal curve and let $(X,D)\to ({\mathbb{P}}^2,E)$ be the minimal log resolution of singularities. We prove that $\bar{E}$ 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 analysing the Minimal Model Program run for the pair $(X,\frac{1}{2}D)$. Namely, we show that ${\mathbb{P}}^2\setminus E$ is ${\mathbb{C}}^{\ast \ast }$-fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

24 pages

References in corpus (2)

• The Coolidge-Nagata conjecture
• The Coolidge-Nagata conjecture, part I

Cited by in corpus (8)

• The Coolidge-Nagata conjecture
• The Coolidge-Nagata conjecture, part I
• Classification of planar rational cuspidal curves. I. C**-fibrations
• Classification of planar rational cuspidal curves. II. Log del Pezzo models
• Involutive Heegaard Floer homology and rational cuspidal curves
• Remark on Tono's theorem about cuspidal curves
• Smooth $\mathbb{Q}$-homology planes satisfying the Negativity Conjecture
• Classification of smooth factorial affine surfaces of Kodaira dimension zero with trivial units