Consider a complete quadrangle $\Delta$ in $\mathbb{CP}^2$ (i.e. the union of the six lines through points $P_1$, $P_2$, $P_3$ and $P_4$ in general position). Let $f: Y := \hat{\mathbb{CP}^2}(P_1, \dots, P_4) \rightarrow \mathbb{CP}^2$ be the Del Pezzo surface of degree $5$. For $j = 1, 2, 3$, let $L'_{j} := H - E_{j} - E_{4}$ and $L_{j} := H - (E_{1} + E_{2} + E_{3}) + E_{j}$, where $H$ is the strict total transform of the line on $\mathbb{CP}^2$, and $E_{1}, \dots, E_{4}$ are the exceptional curves resulting from the blowups. Consider a normal crossing divisor $D := L_1 + L_2 + L_3 + L'_{1} + L'_{2} + L'_{3} + E_{1} + E_{2} + E_{3} + E_{4}$ on $Y$. It is known that there is a smooth Galois cover $p: S \rightarrow Y$ with Galois group $(\mathbb{Z}/5\mathbb{Z})^2$ and with $q = 2$.

I have the following question:


Let $c_3$ and $c_4$ be the generic lines through points $P_3$ and $P_4$ of $\Delta$ (i.e. it do not belong to the branch locus). Let $C_{3}$, $C_{4}$ and $C_{E}$ denote the lifts of the curves $H - E_{3}$, $H - E_{4}$, and $E_{4}$ in $S$. How to compute the intersection numbers $C_{3} \cdot C_{4}$ and $C_{4} \cdot C_{E}$?

It seems to me the curves $C_{4}$ and $C_{E}$, and $C_4$ and $C_3$ will both intersect at five points. I would like to know if there is a formula for computing the intersections of curves in $S$ (using the intersections of curves in $Y$) or for Abelian Galois covering surfaces in general.

I would appreciate any reference.


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