The fundamental locus of a birational map has codimension $2$

Question

Look at the following proposition (Hartshorne Lemma V.5.1):

Let $f:X-\rightarrow Y$ be a birational map between projective varieties, with $X$ normal. Then the locus where $f$ is not defined has codimension $2$ in $X$.

The proof shows the following thing: if $x$ is a point of codimension $1$ (I think that codimension $1$ means that $\mathcal O_{X,x}$ has dimension $1$) then $f$ is defined at $x$.

I don't understand why this fact implies the above proposition! Why if $f$ is defined at points of codimension $1$ then the fundamental locus has codimension $2$?

Answer 1

You should interpret "codimension 2" here as meaning "codimension at least 2".

Of course the fundamental locus could have higher codimension, for example when $f$ is a morphism.

Answer 2

Let $U\subset X$ be the domain of $f$. Suppose $X\setminus U$ has a 1-codimensional irreducible component $V$; the proof shows that the generic point $x$ of $V$ is in $U$, which is a contradiction.