On the projective normality of cyclic coverings over a rational surface

TL;DR

This paper proves that for certain cyclic coverings over rational surfaces, specific adjoint divisors are very ample and normally generated, extending to minimal possibly singular coverings, which advances understanding of their projective embeddings.

Contribution

It establishes new conditions under which adjoint divisors on cyclic coverings over rational surfaces are very ample and normally generated, including cases with singular coverings.

Findings

For any integer k ≥ 3, the divisor K_X + kπ^*A is very ample.

The divisor K_X + kπ^*A is normally generated.

Similar results hold for minimal possibly singular coverings.

Abstract

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 nonruled smooth surface $X$. We show that for any integer $k\ge 3$ and ample divisor $A$ on $S$, the adjoint divisor $K_X+k\pi^*A$ is very ample and normally generated. Similar result holds for minimal (possibly singular) coverings.