Let $X$ be a $3$-fold, and $f:Y\rightarrow X$ a birational $\mathbb{Q}$-factorial divisorial terminal contraction (of relative Picard number one) contracting a divisor $E\subset Y$ to a point $p\in X$. Is the exceptional divisor $E$ necessarily irreducible?



Terminal $\mathbb{Q}$-factorial divisorial contractions
Since $f:Y\rightarrow X$ is terminal we may write
$$K\_Y = f^{\*}K\_X + aE$$
with $a>0$. Let $C\subset Y$ be a curve in the ray contracted by $f$. Then
$$K\_Y\cdot C = C\cdot f^{\*}K\_X+aC\cdot E = K\_X\cdot f\_{\*}C+aC\cdot E = aC\cdot E$$
Now, $K\_Y\cdot C<0$ yields $C\cdot E < 0$.


Assume that $E = E\_1\cup E\_2$ has two components and that $Y$ is $\mathbb{Q}$-factorial. Then $E\_1, E\_2$ are $\mathbb{Q}$-Cartier, $C\cdot E = C\cdot E\_1 + C\cdot E\_2$ implies that $C\cdot E\_1<0$ or $C\cdot E\_2<0$. 


If $C\cdot E\_1<0$ then $C\subset E\_1$ and the curves numerically equivalent to $C$ can not cover $E\_2$ which therefore is not contracted by $f$. 


Similarly, if $C\cdot E\_2<0$ then $C\subset E\_2$ and the curves numerically equivalent to $C$ can not cover $E\_1$ which therefore is not contracted by $f$. 


