The Coolidge--Nagata conjecture

Let E in 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 P^2. The second author recently analyzed the log minimal model program run for the pair (X,1/2 D), where (X,D)→(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 P^2∖E is contracted by the process of minimalization. We prove the conjecture in the remaining cases.