A Lefschetz fibration $M^4\to S^2$ is a generalization of a surface bundle which also allows finitely many nodal singular fibers. The Arakelov–Parshin rigidity theorem implies that holomorphic Lefschetz fibrations of genus $g\ge 2$ admit only finitely many holomorphic sections. In this talk, we will show that no such finiteness result holds for smooth or symplectic sections by giving examples of genus-$g$ ($g\ge 2$) Lefschetz fibrations with infinitely many homologically distinct sections. This is joint work with Carlos A. Serván.