On the projective normality of cyclic coverings over a rational surface

Lei Song

Published 2016-12-20Version 1

Let $S$ be a rational surface with $\dim|-K_S|\ge 1$ and let $\pi: X\rightarrow S$ be a ramified cyclic covering from a smooth surface $X$ with the Kodaira dimension $\kappa(X)\ge 0$. We prove that for any integer $k\ge 3$ and ample divisor $A$ on $S$, the adjoint divisor $K_X+k\pi^*A$ is very ample and normally generated. Similar result holds for minimal (possibly singular) coverings.