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K3 surfaces have been studied from many points of view, but the positivity ofthe cotangent bundle is not well understood. In this paper we explore thesurprisingly rich geometry of the projectivised cotangent bundle of a verygeneral polarised K3 surface $S$ of degree two. In particular, we describe thegeometry of a surface $D_S \subset \mathbb{P}(\Omega_S)$ that plays a similarrole to the surface of bitangents for a quartic in $\mathbb{P}^3$.

The cotangent bundle of K3 surfaces of degree two