# Isomorphisms between complements of projective plane curves

## TL;DR

This paper investigates when complements of irreducible curves in the projective plane are isomorphic, establishing conditions under which such isomorphisms imply projective equivalence, especially for curves of degree up to 7 and providing new examples at degree 8.

## Contribution

It generalizes Yoshihara's result to arbitrary algebraically closed fields and characterizes when complements of certain curves are isomorphic, linking this to projective equivalence.

## Key findings

- Isomorphisms between complements often imply projective equivalence for degree ≤ 7.
- A line intersecting a unicuspidal curve only at its singular point constrains isomorphisms.
- New examples of degree 8 curves with isomorphic complements but non-equivalent are provided.

## Abstract

In this article, we study isomorphisms between complements of irreducible curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if there exists a line that intersects a unicuspidal curve $C \subset \mathbb{P}^2$ only in its singular point, then any other curve whose complement is isomorphic to $\mathbb{P}^2 \setminus C$ must be projectively equivalent to $C$. This generalizes a result of H. Yoshihara who proved this result over the complex numbers. Moreover, we study properties of multiplicity sequences of irreducible curves that imply that any isomorphism between the complements of these curves extends to an automorphism of $\mathbb{P}^2$. Using these results, we show that two irreducible curves of degree $\leq 7$ have isomorphic complements if and only if they are projectively equivalent. Finally, we describe new examples of irreducible projectively non-equivalent curves of degree $8$ that have isomorphic complements.