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Let $X$, $Y$ be connected smooth projective $\mathbb{C}$-schemes. Let $f:Set(X)\rightarrow Set(Y)$ be a bijection of the underlying sets. Suppose that for any $x\in X$, there exists an isomorphism $O_{X, x}\approx O_{Y, f(x)}$. Does there exist an isomorphism $X\rightarrow Y$?

This is true in dimension 1 (the stalk is a field only at the generic point, so the schemes are birational thus isomorphic).

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  • $\begingroup$ Consider $\mathbb A^1$ and $\mathbb P^1$. $\endgroup$
    – Angelo
    Commented Apr 23, 2019 at 4:33
  • $\begingroup$ @Angelo but we require both to be projective $\endgroup$
    – user138661
    Commented Apr 23, 2019 at 4:42
  • $\begingroup$ $\mathbb P^2$ and the blowup of $\mathbb P^2$ at a point provide a projective counterexample. $\endgroup$
    – Angelo
    Commented Apr 23, 2019 at 14:54

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