On the projective normality of cyclic coverings over a rational surface

Abstract

Let $S$ be a rational surface with $\dim |-K_S|\ge 1$ and let $\pi :X\to S$ be a ramified cyclic covering from a nonruled smooth surface $X$. We show that for any integer $k\ge 3$ and ample divisor $A$ 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.

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