Singular Rational Curves on Elliptic K3 Surfaces

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