The Coolidge-Nagata conjecture, part I

TL;DR

This paper proves cases of the Coolidge-Nagata conjecture for rational cuspidal curves in the projective plane using the log minimal model program, establishing conditions under which such curves are Cremona equivalent to a line.

Contribution

It applies the log minimal model program to classify rational cuspidal curves, confirming the conjecture in several new cases based on singularities and geometric properties.

Findings

If $E$ has more than two singular points, then $E$ is Cremona equivalent to a line.

If the divisor $D$ has more than six maximal twigs, then $E$ is Cremona equivalent to a line.

If the complement of $E$ in $\\mathbb{P}^2$ is not of log general type, then $E$ is Cremona equivalent to a line.

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 $D$.