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This question concerns a reduction argument that occurs in the proof of Theorem I.5.3 in Hartshorne. In particular, let $Y$ be an affine variety of dimension $r$ in $\mathbb{A}^n$. Then by (4.9), $Y$ is birational to a hypersurface $X$ of $\mathbb{P}^{r+1}$ (Hartshorne writes $\mathbb{P}^{n}$, I believe this is a typo). Then Hartshorne writes

"Since birational varieties have isomorphic open subsets, we reduce to the case of a hypersurface. It is enough to consider any open affine subset of $Y$, so we may assume that $Y$ is a hypersurface in $\mathbb{A}^n$, defined by a single irreducible polynomial $f(x_1,\dots,x_n) = 0$."

Question 1: I wonder if by the statement "birational varieties have isomorphic open subsets," Hartshorne really means "birational varieties $X, Y$ have open sets $U_X, U_Y$ that are isomorphic" (which is Corollary I.4.5(ii)). The way he writes it, it could mean that for any open set $U_X$ there exists an open set $U_Y$ such that $U_X, U_Y$ are isomorphic. Is this latter statement true? Can somebody please clarify this?

Question 2: Why can we assume that $Y$ is a hypersurface in $\mathbb{A}^{r+1}$ (I believe this is another typo since Hartshorne writes $\mathbb{A}^n$)? Hartshorne's argument seems unclear.

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Q1: As noted in the comments, your interpretation is correct. It's not true that given $X$ birational to $Y$, for every $U_X \subset X$ open there exists an open $U_Y \subset Y$ so that $U_X \simeq U_Y$. Indeed, consider a variety with a point singularity (for instance a nodal cubic) and its desingularization. These varieties are certainly birational, but no open set containing the singularity can be isomorphic to an open subset of the nonsingular variety.

Q2: In the proof of I.5.3, which asserts that the set of singular points of a variety of $Y$ is a proper closed subset, it is first shown that the set of singular points is a closed subset (using affine covers). Reduction to an affine hypersurface is used to show that the set of singular points is a proper subset of $Y$. Note that $Y$ is birational to a hypersurface in $\mathbb P^{r+1}$, and an open subset of this projective hypersurface is isomorphic to a hypersurface in $\mathbb A^{r+1}$. This implies that an open subset $U$ of $Y$ is isomorphic to an open subset of $V$ of an affine hypersurface.

From these observations, it suffices to show that the subset of singular points of the affine hypersurface is a proper subset. Given this, its complement is open by the first part of the proof, and therefore intersects $V$ nontrivially. Since $U \subset Y$ is isomorphic to $V$, it follows that $Y$ contains non-singular points and the set of singular points is a proper subset.

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    $\begingroup$ @Manos, just intersect the hypersurface with an affine open set which intersects it non-trivially. $\endgroup$ – Mariano Suárez-Álvarez Jul 25 '14 at 16:09
  • $\begingroup$ You can choose an appropriate affine open chart for $\mathbb P^n$, and intersect it with your hypersurface. $\endgroup$ – vociferous_rutabaga Jul 25 '14 at 16:10
  • $\begingroup$ Well, try to prove it :-) $\endgroup$ – Mariano Suárez-Álvarez Jul 25 '14 at 16:24
  • $\begingroup$ @Manos you simply need that an open subset of $Y$ is isomorphic to an open subset of an affine hypersurface -- see the clarifications that I've made above. This only requires that some open subset of a projective hypersurface be an affine hypersurface. I've tried to clarify this in my answer. $\endgroup$ – vociferous_rutabaga Jul 25 '14 at 16:33

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