Singular rational curves on elliptic K3 surfaces

Abstract

We show that on every elliptic K3 surface there are rational curves (𝑅 𝑖 ) 𝑖∈ℕ such that 𝑅 2 𝑖 → ∞, that is, of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to ℙ(Ω 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.