Nearby cycle functor for a family of stable curves

Question

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.

Answer 1

Let $\bar M_E = \varepsilon_* \mathbb{Z}$ where $\varepsilon: \tilde E_b\to E_b$ is the normalization map of the special fiber. If the irreducible components $X_1, \ldots, X_r$ of the special fiber are smooth (i.e. have no self-intersections) then $\bar M_E \cong \bigoplus_{i=1}^r \mathbb{Z}_{X_i}$.

Let $\bar M_{E/B}$ be the quotient of $\bar M_E$ by $\mathbb{Z}$ so that we have an extension $$ 0\to \mathbb{Z} \to \bar M_E \to \bar M_{E/B}\to 0. $$

Then there are canonical isomorphisms $$ \mathbb{Z} \cong \psi_f^0(\mathbb{Z}) \quad\text{and}\quad \bar M_{E/B}(-1) \cong \psi_f^1(\mathbb{Z}). $$ Here $(-1)$ means Tate twist i.e. tensor product with $\frac{1}{2\pi i} \mathbb{Z}$.

In particular, the local system $\psi_f^0(\mathbb{Z})$ is constant, and $\psi_f^1(\mathbb{Z})$ is just (but noncanonically) a copy of $\mathbb{Z}$ at each node.

EDIT. Some references: This goes back at least to SGA7 where nearby cycle complexes were originally defined. I highly recommend Illusie's survey Autour du theoreme de monodromie locale in Asterisque 223 (in particular, section 2.1), although it is much more general than what you need.. Later it was realized by Kato, Nakayama, and others that one can explicate in a very simple way nearby cycles in a semistable situation using logarithmic geometry (the notation $\bar M_E$ I used above was motivated by this). For this, there is another nice survey by Illusie in another Asterisque. Also see 4.9 of this.