 a generic Weil divisor on $(X,P)$ of class $k$, that is, a generic germ in $\cO_{X,\zeta^k}$. In [@jiJM-correction] it is shown that $$A_{X,P}(D)=\delta^{\operatorname{top}}(D)-\kappa_P(D).$$ When applied this formula for generic germs one obtains a combinatorial way to calculate $A_{X,P}(D)$, that is, $$A_{X,P}(D)=\delta^{\operatorname{top}}(D)-r_P(D)+1.$$

Cyclic coverings of smooth algebraic surfaces d’après Esnault-Viehweg
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Esnault-Viehweg’s theory for cyclic covers in the smooth case can be presented as follows. Consider $X$ a projective smooth surface and let $D$ be a divisor which is linearly equivalent to $nH$ where $H$ is another divisor. Then $\mathcal{O}_X(D)$ is isomorphic to $\mathcal{O}_X(H)^{\otimes n}$. Then, given a meromorphic section $t:X\dashrightarrow\mathcal{O}_X(D)$, such that $\operatorname{div}(t)=D$, we can consider $$\hat{X}:=\overline{\{(x,v)\in\mathcal{O}_X(H)\mid v^{\otimes n}=t(x)\}}$$ and a suitable smooth model $\pi:\tilde{X}\to \hat X$. Recall that the first Betti number of $\tilde{X}$ equals $2\dim H^1(\tilde{X},\mathcal{O}_{\tilde{X}})$ (its irregularity) and it is a birational invariant.

Let us assume now that $D$ is a simple normal crossing divisor. We can assume that $D$ is effective and $D=\sum_{j=1}^r n_i D_i$ is its decomposition in irreducible components and $0\leq n_i<n$ (note that $n_i=0$ means that the covering is not ramified along $D_i$, but we allow this for technical reasons). Using the eigen-decomposition of $\pi_*(\mathcal{O}_{\tilde{X}})$ induced by that of $\mathcal{O}_{\tilde{X}}$ one can describe the irregularity of $\tilde X$ in cohomological terms using line bundles on $X$.

Under the previous conditions, the irregularity of the covering $\tilde{X}$ equals $2\sum_{k=0}^{n-1}\dim H^1(X,\mathcal{L}^{(k)})$, where $$\label{eq:Lkliso} 
\mathcal{L}^{(k)}=
\mathcal{O}_X
\left(
-k H+
\sum_{i=0}^r
\left\lfloor\frac{k n_i}{n}\right\rfloor 
D_i
\right).$$

One of the goals of this paper