Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection and simple normal crossings such that there is a morphism $f:S\rightarrow S'$ which collapses them to a point.
Can we form an effective divisor $D=\sum d_i D_i$ such that $D\cdot D_i<0$ for all $i$? Note that if I had said $\leq$ rather than $\lt$ this is the content of the Proposition in page 83 of Reid's Park City lectures. I ask for strict inequality.
My belief is that this is possible, but I don't manage to give a proof nor find a reference or even to find a counter-example... This would be very useful for many arguments in birational geometry of surfaces.