  Let $S$ be a smooth projective complex algebraic surface and $f\, :\, S\,
\longrightarrow\, {\mathbb C}{\mathbb P}^2$ a finite map. Consider a pencil of
hyperplane sections on ${\mathbb C}{\mathbb P}^2$ and pull it back to $S$. We
address the question whether such a pencil is a Lefschetz pencil on $S$.
