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?

## 1 Answer

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 $f$.