Are pseudo-isomorphism between normal surfaces isomorphisms?

Question

Let $X,Y$ be two normal algebraic surfaces (for instance projective) and let $\varphi\colon X\dashrightarrow Y$ be a birational map which restricts to an isomorphism $(X\setminus F)\to (Y\setminus G)$ where $F\subset X,G\subset Y$ are finite subsets. Does it follow that $\varphi$ is an isomorphism?

(This is true at least when $X$ and $Y$ are smooth).

Answer 1

I think so (even in dimension higher than 2, assuming that $F$ and $G$ are still finite sets, and not codimension two subvarieties of course). Let $H$ be an ample divisor in $X$ avoiding $F$ and let $H'$ be its strict transform in $Y$. Then $H'$ is ample in $Y$. In particular $Y={\rm Proj}(Y,H')={\rm Proj}(X,H)=X$.