# Terminal $\mathbb{Q}$-factorial divisorial contractions

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?

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 $$f$$.