New distinct curves having the same complement in the projective plane

TL;DR

This paper constructs new unicuspidal plane curves of degree 4m+1 that serve as counterexamples to Yoshihara's conjecture, showing that isomorphic complements do not imply projective equivalence.

Contribution

It introduces a new family of unicuspidal curves of degree 4m+1 that counter the Yoshihara conjecture, including the lowest known degree counterexamples.

Findings

Counterexamples of degree 9 to Yoshihara's conjecture.

All counterexamples are unicuspidal curves.

The degree of these curves is 4m+1 for m ≥ 2.

Abstract

In 1984, H. Yoshihara conjectured that if two plane irreducible curves have isomorphic complements, they are projectively equivalent, and proved the conjecture for a special family of unicuspidal curves. Recently, J. Blanc gave counterexamples of degree 39 to this conjecture, but none of these is unicuspidal. In this text, we give a new family of counterexamples to the conjecture, all of them being unicuspidal, of degree 4m + 1 for any m \geq 2. In particular, we have counterexamples of degree 9, which seems to be the lowest possible degree.