Exceptional locus of a projective birational morphism between smooth varieties Let $X \rightarrow Y$ be a birational projective morphism between smooth varieties over $\mathbb{C}$. 
I think that the exceptional locus $E \subset X$ of $f$ is codimension $1$. 
Assume that $\dim X = \dim Y =3$. 
Question Is $E$ normal crossing? 
If you know counterexample, please let me know. 
 A: The answer is no.  Here's an example.
Fix $Y$ and a closed point $p \in Y$.  Blow up $p$.  This is clearly smooth.  Call the resulting scheme $X'$ and let $F$ denote the exceptional divisor of $f' : X' \to Y$.  I am actually going to assume that $Y = \mathbb{A^3}$ in my mind.
Here's the idea then, within $X'$, choose a smooth curve $C$ which is tangent to $F$ at some point $p'$ in some "sufficiently funny way" and blow up the entire curve $C$.  
Explicitly, if $X'$ has a chart $\text{Spec} k[x,y,z]$ and $F$ is given by the equation $z = 0$, then you can set $C = V(y, x^2 + z)$.  This gives us $g : X \to X'$.  Certainly $X$ is smooth and $f : X \xrightarrow{g} X' \xrightarrow{f'} Y$ is birational.  
Now, I claim that the exceptional divisor of $f$ is not simple normal crossing.  It is enough to show that the strict transform of $F$ in $X$ is not smooth.  In our particular example, the strict transform of $F$ corresponds to the blowup of $(y,x^2)$ within $k[x,y]$.  It is therefore easy to see that the strict transform of $F$ has a quadric cone singularity.
