Let $B$ be a smooth algebraic curve over $\mathbb{C}$ (or rather a germ of it at a point $b\in B$). Let $f\colon E\to B$ be a proper flat family of stable curves with smooth generic fiber. Assume that the fiber $f^{-1}(b)$ is not smooth.

Let us consider the nearby cycle functor $\psi_f(\underline{\mathbb{C}})$ of the constant sheaf on $E\backslash f^{-1}(b)$ which is a complex of sheaves on the special fiber $f^{-1}(b)$. Outside of singular points it is a local system of rank 1. If I understand correctly, at singular points $\psi_f(\underline{\mathbb{C}})$ is concentrated at two degrees, 0 and 1 (after appropriate normalization). I expect that the sheaf cohomology in degree 0 has rank 1 (am I wrong?). Let us denote by $r$ the rank of the sheaf cohomology in degree 1.

**Question.** What are the possible values of $r$?

In a simple example I know $r=1$, but can one have $r>1$?

I apologize if this question is not of a research level, I am not an expert in the field.