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, that is, it is mapped onto a line by some birational transformation of ${\mathbb{P}}^2$. The second author recently 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 proved the conjecture in the 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.