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From Mumford's Red Book, Chapter 2, Example K:

Take $X = Y = \mathbb{P}^2$, and let $x_0, x_1, x_2$ and $y_0, y_1, y_2$ be homogeneous coordinates on $X$ and $Y$. Let $U_0 \subset X$ and $V_0 \subset Y$ be defined as the open sets $x_0 x_1 x_2 \neq 0$ and $y_0 y_1 y_2 \neq 0$. Define an isomorphism between $U_0$ and $V_0$ by the map $y_i = 1/ x_i$. In fact, this is just an extension of the isomorphism of function fields:

$\begin{align} k\left(\frac{x_1}{x_0}, \frac{x_2}{x_0}\right) &\tilde\rightarrow k\left(\frac{y_1}{y_0}, \frac{y_2}{y_0}\right) \\\\ x_1/x_0 &\mapsto y_0/y_1 \\\\ x_2/x_0 &\mapsto y_0/y_2 \end{align}$

I'm confused as to what is meant by "an extension" here -- is this not just the map induced by this map of function fields? What am I missing?

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If $X$ and $Y$ are two varieties, then any isomorphism $K(X) \cong K(Y)$ arises from an isomorphism $U \cong V$ of non-empty open subsets $U$ and $V$ of $X$ and $Y$, but it need not extend to an isomorphism $X \cong Y$ (although if it does, the extension is unique). In this case it does so extend, and Mumford is remarking on that fact.

(The $X$ and $Y$ of my discussion are Mumford's $U_0$ and $V_0$; sorry for the notational clash.)

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