On the projective normality of cyclic coverings over a rational surface

Let $S$ be a rational surface with $\dim \mathrm{\mid }-K_S\mathrm{\mid }\ge 1$ and let $\pi: X\rightarrow S$bearamifiedcycliccoveringfromanonruledsmoothsurface$X$\mathrm{.}Weshowthatforanyinteger$k\ge 3$andampledivisor$A$on$S$, the adjoint divisor $K_X+k{\pi }^{\ast }A$ is very ample and normally generated. Similar result holds for minimal (possibly singular) coverings.