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.

Export event as iCal