The Coolidge-Nagata conjecture (Q1681618)

The Coolidge-Nagata conjecture asserts that any rational cuspidal curve of the complex projective plane is the image of a line by a Cremona transformation. This paper completes the proof of the conjecture that was started in [\textit{K. Palka}, Adv. Math. 267, 1--43 (2014, Zbl 1327.14153)]. The proof proceeds by applying the log minimal model program to the pair \((X, \frac12 D)\), where \((X,D) \to (\mathbb P^2,E)\) is a minimal resolution of the singularities of the cuspidal curve \(E\). The second author introduced the concept of ``almost minimal model'', and proved the conjecture when the number \(n\) of steps before reaching such a model is greater than 1. The present paper treats the remaining cases \(n = 1\) or \(0\).