# Are singular rational curves on K3 surfaces rigid?

Let $S$ be a K3 surface over the complex numbers $\mathbb{C}$. If $C\subset S$ is a smooth rational curve, the normal bundle $N_{C/S}$ is isomorphic to $\mathbb{O}(-2)$ and thus $C$ is rigid. What about a curve $C'\subset S$ whose normalization is a rational curve? I think it may admit deformation in higher genus family, but can it admit deformation in a family of (possibly singular) rational curves?

• Not over $\mathbb{C}$, but this can occur in positive characteristic. – Jason Starr Dec 18 '13 at 15:02

To develop what Jason says: if your curve deforms in a family of rational curves, it means that you can find a dominant rational map from a ruled surface onto your K3. This is forbidden (over $\mathbb{C}$): e.g. because the nonzero 2-form of the K3 would lift to a nonzero 2-form on the ruled surface.