 Euler characteristic of a Lefschetz pencil

If a Lefschetz pencil on a closed four-manifold $X$ has smooth fiber of genus $g$, $b$ base points and $m$ critical points, thenIndeed, blowing up the base points gives $\chi (\tilde{X})=\chi (X)+b$, while a genus-$g$ fibration over $S^2$ contributes $2(2-2g)$ and each Lefschetz critical point contributes one.