Abstract: This is joint work with Jérémy Blanc and Pierre-Marie Poloni. In this talk we will focus on the complement problem for the projective space P^n: If H, H’ are irreducible hypersurfaces of degree d in P^n such that the complements P^n\H and P^n\H are isomorphic, are the hypersurfaces H, H’ isomorphic?

This problem has been studied intensively for n=2 and we will focus in this talk on the case n > 2. In fact, in our main result, we provide counterexamples for all n, d > 2 provided that (n, d) is not equal to (3, 3) and give partial affirmative answers in case (n, d) = (3, 3) and d < 3. As a byproduct, we show that rational normal projective surfaces admitting a desingularisation by trees of smooth rational curves are piecewise isomorphic if and only if they coincide in the Grothendieck ring.