Seminar room 00.003

## Seminar Algebra and Geometry: Mattias Hemmig (Basel)

**Isomorphisms between complements of unicuspidal curves in the projective plane**

In 2012 Costa constructed a family of unicuspidal curves of degree 9 in the projective plane that are pairwise non-equivalent but have isomorphic complements. We show that a family as the one of Costa cannot contain a unicuspidal curve *C* that admits a line intersecting *C* only in the singular point. To state the result more precisely, if *D* is any plane curve and there exists an isomorphism between the complements of *C* and *D*, then the two curves are projectively equivalent, even though the isomorphism is not necessarily linear. The proof works over an algebraically closed field of any characteristic and generalizes a result of Yoshihara (1984) who proved the claim over the complex numbers. We then use this result to show that any two irreducible curves of degree at most 8 have isomorphic complements if and only if they are projectively equivalent.

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