Isomorphisms between complements of projective plane curves
| Autor: | Mattias Hemmig |
|---|---|
| Jazyk: | English<br />French |
| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Épijournal de Géométrie Algébrique, Vol Volume 3 (2019) |
| Druh dokumentu: | article |
| ISSN: | 2491-6765 03411990 |
| DOI: | 10.46298/epiga.2019.volume3.5541 |
| Popis: | 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. |
| Databáze: | Directory of Open Access Journals |
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