Hartshorne Theorem I.5.3

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.

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.
• You can choose an appropriate affine open chart for $\mathbb P^n$, and intersect it with your hypersurface. – vociferous_rutabaga Jul 25 '14 at 16:10
• @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. – vociferous_rutabaga Jul 25 '14 at 16:33