Unboundedness of fixed point multiplicities on a K3 surface

TL;DR

This paper demonstrates that on a specific K3 surface, automorphisms can have fixed points with arbitrarily large multiplicities, challenging traditional bounds and revealing new geometric behaviors.

Contribution

It constructs explicit automorphisms with fixed points of unbounded multiplicity and shows intersection multiplicities can also be arbitrarily large on this K3 surface.

Findings

Automorphisms with fixed points of arbitrarily large multiplicity.

Intersection multiplicity of isomorphic curves can be arbitrarily large.

Automorphisms can be arbitrarily close to the identity at fixed points.

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.