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arXiv · 1405.5346

Cuspidal curves, minimal models and Zaidenberg's finiteness conjecture

Karol Palka
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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 $\mathbb{C}^{**}$-fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

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Karol Palka. 2014-05-21. Cuspidal curves, minimal models and Zaidenberg's finiteness conjecture. https://doi.org/10.1515/crelle-2016-0021

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