Normal generation of special line bundles on multiple coverings

Abstract

Let X be a smooth curve of genus g which is a multiple covering $\varphi :X\to Y$ of a smooth curve Y. We define an invariant ${\Delta }_{\varphi }$ of the map $\varphi$, which is the maximum among the integers m such that for a base point free line bundle $\mathcal{L}$ on X with $\mathrm{\text{deg}}\mathcal{L}\le g-1$ the inequality $\text{Cliff}(\mathcal{L})\le m$ implies that $\mathcal{L}$ is composed with $\varphi$. Our main result is that a very ample special line bundle $\mathcal{L}$ on X with $\mathrm{\text{deg}}\mathcal{L}>\frac{3g-3}{2}$ is normally generated when $\text{Cliff}(\mathcal{L})\le {\Delta }_{\varphi }$ under additional numerical conditions.

Keywords:

• Clifford index
• multiple covering
• normal generation
• special line bundle

2020 Mathematics Subject Classification:

• 14C20
• 14H51
• 14H45

Acknowledgments

The authors would like to thank Korea Institute for Advanced Study (KIAS) for the warm hospitality when they were visiting there. We also thank the referee for his/her useful comments on our manuscript. Especially we extend the bound of $\gamma$ in Lemma 2.6 and in Theorem 3.1 thanks to the referee’s comments.

Disclosure statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Funding

This work was supported by the 2023 Yeungnam University Research Grant.