On the rectifiability of rational plane curves

Let $\bar E\subseteq \mathbb{P}^2$ be a rational cuspidal curve defined over complex numbers. The Coolidge-Nagata conjecture states that such a curve is rectifiable, i.e. it can be transformed into a line by a birational automorphism of $\mathbb{P}^2$. We will prove some new results in this direction, showing in particular that the conjecture holds if $\bar E$ has more than four cusps.