Let $k$ be an algebraically closed field. Let $H_1, H_2$ be two smooth hypersurfaces of the same degree $d$ in $P^n_k$. Let $U_1,U_2$ be their complements respectively. Are $U_1,U_2$ isomorphic as algebraic varieties?
In $n=1,d=1$ case this is true, because the complement of any point is isomorphic to $A^1$.
But $n=2$ case I guess this might be false. I want to prove that if $U_1,U_2$ are isomorphic then they must be induced by an automorphism of $P^n$, but this seems hard. I read something about the 'complement problem' on enter link description here, but this seems to be a more complicated question, and it focuses on $A^n$ instead.
Maybe the $n=2,d=3$ case is easier? In this case, elliptic curves are isomorphic if and only if they have the same $j$-invariant. Can we read this from its complement?
Are there any solutions/counterexamples? Any comments are welcome!