Suppose ${\Bbb C}$ be a complex number field and $S$ be a projective smooth surface over ${\Bbb C}$. We consider a finite etale Galois covering $\pi \colon T \to S$ of degree $d$. We consider the curve $D$ on $S$ such that
$D = \underset{i = 1,\cdots,n}{\cup} C_i$, where each $C_i$ is a smooth irreducible curve and $C_i \cdot C_j = 1$ for $i \not= j$.
$D$ forms a closed loop, i.e. the dual graph of $D$ is a $n$-gon.
Each irreducible component $C_i$ of $D$ splits into $d$ irreducible components in $T$.
$D$ can be contracted to a point. I.e., $\exists f \colon S \to U$ such that $U$ is a proper algebraic surface over ${\Bbb C}$ and $f(D) = {\mathrm{a~point}}$.
Q. For a given etale covering $\pi$, is it true that sufficiently many $D$'s satisfying above 4 conditions exist in quantity? Or is there any measure to size up the number of above $D$'s?