Let $X,Y$ are nonsingular complex projective surfaces and $f:Y\to X$ be a birational morphism. Let $E_1,\ldots,E_k$ are irreducible exceptional divisors of $f$.
In this situation, we have $K_Y=f^{\ast}K_X+\sum_{i=1}^{k}m_iE_i (m_i>0)$ ($K_Y,K_X$: canonical divisor of $Y,X$)
How to prove $(f^{\ast}K_X)\cdot E_i=0 (\forall i)$?
More generally,Is it true that $(f^\ast D)\cdot E_i=0$ for all $i$ and for all divisor $D$ on $X$?