I have the following Lemma (http://www.jmilne.org/math/xnotes/AVs.pdf, Lemma 3.2):
A rational map $f:V\dashrightarrow W$ from a normal variety to a complete variety is defined on an open subset $U$ of $V$ whose complement $V\setminus U$ has codimension $\geq 2$
PROOF: Let $v\in V\setminus U$ be a point of codimension $1$ in $V$. Then $\mathcal{O}_{V,v}$ is a discrete valuation ring with field of fractions $k(V)$ because $V$ is normal. The valuative criterion of properness shows that the map Spec$(k(V)) \rightarrow W$ defined by $f$ extends to a map Spec$(\mathcal{O}_{V,v}) \rightarrow W$. This implies that $f$ has a representative defined on a neighborhood of $v$ and so $v\in U$. This proves the lemma.
I have got two questions:
- How can I find the map Spec$(k(V)) \rightarrow W$?
- How can I conclude that $f$ has a representative defined on a heighborhood of $v$?
For the first question, look at the generic point of $V$. Let $\eta_V$ be the generic point. $\eta_V=\mathrm{Spec} K(V)\hookrightarrow V$ naturally, and then compose with the map $V\stackrel{f}{\to} W$ to get a map $\mathrm{Spec} K(V)\to W$.
For the second, go back to Hartshorne and draw the correct square from the valuative criteria for properness. Vertically, your arrows should be $\mathrm{Spec}(K(V))\to \mathrm{Spec} \mathcal{O}_{V,v}$ and $W \to \mathrm{Spec} k$, the structure morphism. Horizontally, the top arrow $\mathrm{Spec} K(V)\to W$ is the one from the first paragraph, and the bottom arrow $\mathrm{Spec}\mathcal{O}_{V,v}\to \mathrm{Spec} k$ is the structure morphism. Since a complete variety is proper, there exists a unique map $\mathrm{Spec} \mathcal{O}_{V,v}\to W$ making the square commute. This map is defined on some open neighborhood of $V$ and it agrees with $f$ there, so we have what we want.