The Coolidge-Nagata conjecture

TL;DR

This paper proves the Coolidge-Nagata conjecture, which states that complex rational cuspidal curves in the projective plane are Cremona equivalent to a line, completing previous partial results by analyzing the minimal model program.

Contribution

The paper completes the proof of the Coolidge-Nagata conjecture by addressing cases not covered in prior work, confirming that all such curves are Cremona equivalent to a line.

Findings

Confirmed the conjecture for all rational cuspidal curves in the plane.

Extended previous results by analyzing remaining cases in the minimal model program.

Established that these curves are always Cremona equivalent to a line.

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.