ag.algebraic geometry - Negativity of contraction

Dear Carlos, the statement is false in general. For example let $Y$ be $mathbb{C}^3$, let $f_1 : X_1 rightarrow Y$ be the blowup of a point on $Y$, and $f_2 : X rightarrow X_1$ the blowup of a point on the exceptional divisor of $f_1$. Let $f : X rightarrow Y$ be the composition. The exceptional locus of $f$ has two components: a copy $F$ of $mathbb{P}^2$ (the exceptional divisor of $f_2$) and a copy $E$ of the blowup of $mathbb{P}^2$ in one point (the strict transform of the exceptional divisor of $f_1$). The intersection $C = E cap F$ is a copy of $mathbb{P}^1$, it is a line on $F$ and the $f_2$-exceptional curve on $E$. Now $E cdot C$ equals $C^2$ computed on $F$ (because $C=E cap F$),
so $E cdot C = +1$.