A famous theorem of Castelnuovo and de Franchis tells us that for $S$ a smooth projective complex algebraic surface that for $b \geq 2$, pencils $f : S \to B$ of genus $b :=g(B)$ are in bijective correspondence with subspaces $W \subset H^0(S, \Omega_S^1)$ of dimension $b$ such that $\bigwedge^2W = 0$ and $W$ is maximal with this property. In particular, we know by work of Beauville that the property that a surface has a pencil of genus at least $b$ is topological: such a pencil exists if and only if $\pi_1(S) \twoheadrightarrow \pi_1(S_b)$ where $S_b$ is a Riemann surface of genus $b$. See the article of Mendes Lopes and Pardini for more details.
I'm interested in the question of whether $S$ has more than one such pencil. Is there a topological characterization of this? Are there numerical constraints on $S$ that we can impose that forces this to happen?
The only theorem I could find relating to this question is a classical theorem of Severi quoted in the above-linked article saying that there are finitely many pencils of genus $\geq 2$ for a fixed surface $S$.
