Isomorphisms between complements of plane curves

We study isomorphisms between complements of irreducible plane curves. In the first part, we give a new counterexample to the complement problem in the projective plane (Yoshihara's conjecture) of degree 8 and show that there exist no counterexamples of lower degree. Moreover, we show that Yoshihara's conjecture holds for unicuspidal curves that admit a very tangent line through the singular point. This generalizes a result of Yoshihara, who proved this claim over the complex numbers. In the second part, coauthored with Jérémy Blanc and Jean-Philippe Furter, we give the first counterexample to the complement problem in the affine plane. Moreover, we prove that any isomorphism between the complements of two irreducible affine plane curves extends to an automorphism of the affine plane if the curves are not isomorphic to open subsets of the affine line. Finally, in the third part, we study a conjecture of Sathaye concerning lines in the affine plane and show that this conjecture holds for curves up to degree 11.