S(\lambda_2,\ldots, \lambda_g)$ under the blow-up $\sigma$, then our interest goes to the monodromy zeta function $Z_{Y,p}^{mon}(t)$ at the point $p := S' \cap \sigma^{-1}(0)$. Using the above A’Campo’s formulas, it turns out that, for generic coefficients, $Z_{Y,p}^{mon}(t)$ is equal to the monodromy zeta function $Z_{Y,0}^{mon}(t)$ of $Y$ considered on $S$ at the origin; this will be shown in Theorem \[thm:red-to-on-curve-surface\]. In fact, this result will be stated and proven in a more general context, which makes it possibly useful for other instances of the monodromy conjecture.

To compute the monodromy zeta function $Z_{Y,0}^{mon}(t)$ of $Y \subset S$ at the origin, we will consider another generalization of A’Campo’s formula in terms of an *embedded ${\mathbb Q}$-resolution* of $Y \subset S$ that was proven in [@Ma1]. Roughly speaking, a ${\mathbb Q}$-resolution is a resolution in which the final ambient space is allowed to have abelian quotient singularities, and the zeta function of monodromy at the origin can be written as $$Z^{mon}_{Y,0}(t) = \prod_{\underset{t \in T}{1 \leq j \leq r}} \left(1-t^{m_{j,t}}\right)^{\chi(E_{j,t}^{\circ})},$$ where $\{E_{j,t}\}_{j=1,\ldots,r,t \in T}$ defines a finite stratification of the exceptional varieties $E_1,\ldots,E_r$ of the ${\mathbb Q}$-resolution such that the *multiplicity* $m_{j,s}$ of $E_j$ along each $E_{j,t}$ is constant. To construct an 