  Let E be a plane rational curve defined over complex numbers which has only
locally irreducible singularities. The Coolidge-Nagata conjecture states that E
is rectifiable, i.e. it can be transformed into a line by a birational
automorphism of the plane. We show that if it is not rectifiable then the tree
of the exceptional divisor for its minimal embedded resolution of singularities
has at most nine maximal twigs. This settles the conjecture in case E has more
than four singular points.
