The correspondence between a plane curve and its complement

TL;DR

This paper investigates whether plane curves with isomorphic complements are related by plane automorphisms, providing counterexamples to a conjecture and revealing unexpected automorphisms of affine surfaces.

Contribution

It constructs counterexamples to Yoshihara's conjecture, showing that isomorphic complements do not always imply automorphic equivalence of curves.

Findings

Counterexamples to Yoshihara's conjecture over any ground field

Existence of non-linear automorphisms of affine surfaces

Curves can be isomorphic or non-isomorphic in the counterexamples

Abstract

Given two irreducible curves of the plane which have isomorphic complements, it is natural to ask whether there exists an automorphism of the plane that sends one curve on the other. This question has a positive answer for a large family of curves and H.Yoshihara conjectured that it is true in general. We exhibit counterexamples to this conjecture, over any ground field. In some of the cases, the curves are isomorphic and in others not; this provides counterexamples of two different kinds. Finally, we use our construction to find the existence of surprising non-linear automorphisms of affine surfaces.