Quadrics and normal generation of line bundles on multiple coverings

Abstract

In this work, a minimally FIIQ divisor on a smooth curve $X$ in ${\mathbb{P}}^r$ is defined by an effective divisor which fails to impose independent conditions on quadrics in ${\mathbb{P}}^r$ and all of whose proper subdivisors impose independent conditions on quadrics. Using this notion, we explicitly describe very ample non-special line bundles which fail to be normally generated on a simple $k$-fold covering $\varphi :X\to Y$ in terms of line bundles on $Y$ under some numerical constraints. Furthermore, in case $g_Y=1,2$ we obtain a concrete classification of such line bundles on $X.$

Keywords:

• Multiple covering
• non-special line bundle
• normal generation
• quadrics

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 14C20
• 14H51
• 14H45
• 14H30

Acknowledgement

The authors would like to thank Korea Institute for Advanced Study (KIAS) for the warm hospitality when they were visiting there.

Additional information

Funding

The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1I1A3A01055643). The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2019R1F1A1058248).