On the projective normality of cyclic coverings over a rational surface

Let S be a rational surface with dim|−KS|⩾1 and let π:X→S be a ramified cyclic covering from a nonruled smooth surface X . We show that for any integer k⩾3 and ample divisor A on S , the adjoint divisor KX+kπ∗A is very ample and normally generated. Similar result holds for minimal (possibly singular) coverings.