The Coolidge-Nagata conjecture, part I

Abstract

Let $E\subseteq {\mathbb{P}}^2$ be a complex rational cuspidal curve contained in the projective plane and let $(X,D)\to ({\mathbb{P}}^2,E)$ be the minimal log resolution of singularities. Applying the log minimal model program to $(X,\frac{1}{2}D)$ we prove that if $E$ has more than two singular points or if $D$, which is a tree of rational curves, has more than six maximal twigs or if ${\mathbb{P}}^2\setminus E$ is not of log general type then $E$ is Cremona equivalent to a line, i.e. the Coolidge-Nagata conjecture for $E$ holds. We show also that if $E$ is not Cremona equivalent to a line then the morphism onto the minimal model contracts at most one irreducible curve not contained in $D$.

34 pages, 1 figure

References in corpus (2)

• Cuspidal curves, minimal models and Zaidenberg's finiteness conjecture
• The Coolidge-Nagata conjecture holds for curves with more than four cusps

Cited by in corpus (5)

• The Coolidge-Nagata conjecture
• 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
• Involutive Heegaard Floer homology and rational cuspidal curves