The Coolidge-Nagata conjecture, part I

Let E⊆P2 be a complex rational cuspidal curve contained in the projective plane and let (X,D)→(P2,E) be the minimal log resolution of singularities. Applying the log Minimal Model Program to (X,12D) 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 P2∖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.