Nonlinearizable embeddings of elliptic curves in rational surfaces

We show that for any smooth cubic in ${\mathbb{P}}^2$, there exists a dense $G_{\delta }$ set of configurations of 9 distinct points such that blowing up ${\mathbb{P}}^2$ at these 9 points, the strict transform of the cubic is not linearizable and has nontorsion normal bundle. This answers a problem raised by Ogus in 1975.