Unboundedness of fixed point multiplicities on a K3 surface

Abstract

We exhibit automorphisms of a certain K3 surface in $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$ with an isolated fixed point at which the induced action on the stalk of the structure sheaf is arbitrarily close to the identity. This implies that the multiplicities of these automorphisms at the fixed point can be arbitrarily large. As another application, we show that the intersection multiplicity of two isomorphic curves at a point can be arbitrarily large on this K3 surface.