Mattias Hemmig - Isomorphisms between complements of projective plane curves

epiga:5541 - Épijournal de Géométrie Algébrique, November 13, 2019, Volume 3 -
Isomorphisms between complements of projective plane curvesArticle

Authors: Mattias Hemmig

    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.

    Volume: Volume 3
    Published on: November 13, 2019
    Accepted on: September 24, 2019
    Submitted on: June 3, 2019
    Keywords: Mathematics - Algebraic Geometry,14E07, 14H45, 14H50, 14J26

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