I}$ at $e \in E'$ as $$Z_{{\mathcal I},e}^{mon}(t) := \prod_{k \geq 0}\det(\text{Id} - t\tilde{M}^k_y)^{(-1)^k},$$ where $y \in C_YX\setminus Y$ is an arbitrary point in $p^{-1}(e)$. Denef’s result [@Den2 Lemma 4.6] can be generalized to this situation stating that every monodromy eigenvalue associated with ${\mathcal I}$ is a zero or pole of the monodromy zeta function of ${\mathcal I}$ at some point $e \in E'$.

[@VV] \[thm:ACampo-Princ\] Let ${\mathcal I}$ be a sheaf of ideals on a variety $X$. Let $Y = V({\mathcal I})$ be the associated subscheme in $X$ and suppose that $\text{Sing}(X) \subset Y$. Consider a principalization $\varphi: \tilde X \rightarrow X$ of ${\mathcal I}$. Denote by $E_j$ for $j \in J$ the irreducible components of $\varphi^{-1}(Y)$ with numerical data $(N_j,\nu_j)$ and define $E^{\circ}_j = E_j \setminus \cup_{i \neq j}(E_i \cap E_j)$ for every $j \in J$. Let $\sigma: X' \rightarrow X$ be the blow-up of $X$ with center $Y$ and let $E' = \sigma^{-1}(Y)$ be its exceptional divisor. By the universal property of the blow-up, there exists a unique morphism $\psi: \tilde X \rightarrow X'$ such that $\sigma \circ \psi = \varphi$. For a point $e \in E'$, the zeta function of monodromy of ${\mathcal I}$ at $e$ 