The Coolidge-Nagata conjecture

Abstract

Let $E\subseteq {\mathbb{P}}^2$ be a complex rational cuspidal curve contained in the projective plane. The Coolidge-Nagata conjecture asserts that $E$ is Cremona equivalent to a line, i.e. it is mapped onto a line by some birational transformation of ${\mathbb{P}}^2$. In arXiv:1405.5917 the second author analyzed the log minimal model program run for the pair $(X,\frac{1}{2}D)$, where $(X,D)\to ({\mathbb{P}}^2,E)$ is a minimal resolution of singularities, and as a corollary he established the conjecture in case when more than one irreducible curve in ${\mathbb{P}}^2\setminus E$ is contracted by the process of minimalization. We prove the conjecture in the remaining cases.

38 pages, 5 figures

References in corpus (5)

• Cuspidal curves, minimal models and Zaidenberg's finiteness conjecture
• The Coolidge-Nagata conjecture, part I
• A new proof of the theorems of Lin-Zaidenberg and Abhyankar-Moh-Suzuki
• On a new class of rational cuspidal plane curves with two cusps
• The Coolidge-Nagata conjecture holds for curves with more than four cusps

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• Cuspidal curves, minimal models and Zaidenberg's finiteness conjecture
• Classification of planar rational cuspidal curves. I. C**-fibrations
• Classification of planar rational cuspidal curves. II. Log del Pezzo models
• The 2-Hessian and sextactic points on plane algebraic curves
• Involutive Heegaard Floer homology and rational cuspidal curves
• Dimension counts for singular rational curves via semigroups
• Topological obstructions for rational cuspidal curves in Hirzebruch surfaces
• Smooth $\mathbb{Q}$-homology planes satisfying the Negativity Conjecture