Singular rational curves on elliptic K3 surfaces

We show that on every elliptic K3 surface there are rational curves (Ri)i∈N$(R_i)_{i\in \mathbb {N}}$ such that Ri2→∞$R_i^2 \rightarrow \infty$, that is, of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to P(ΩX)$\mathbb {P}(\Omega _X)$ is dense in the Zariski topology. As an application, we give a simple proof of a theorem of Kobayashi in the elliptic case, that is, there are no globally defined symmetric differential forms.