Abstract
In 1984, 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, 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 ≥ 2. In particular, we have counterexamples of degree 9, which seems to be the lowest possible degree.
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References
Blanc J.: The correspondance between a plane curve and its complement. J. Reine Angew. Math. 633, 1–10 (2009)
Yoshihara H.: On open algebraic surfaces \({\mathbb{P}^{2} - C}\) . Math. Ann. 268, 43–57 (1984)
Yoshihara H.: Rational curves with one cusp II. Proc. Am. Math. Soc. 100, 405–406 (1987)
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Costa, P. New distinct curves having the same complement in the projective plane. Math. Z. 271, 1185–1191 (2012). https://doi.org/10.1007/s00209-011-0909-4
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DOI: https://doi.org/10.1007/s00209-011-0909-4