Rank two ACM bundles on the double embedding of a quadric surface

Abstract

In this article, we completely classify locally free sheaves $\mathcal{E}$ of rank 2 such that $h^1(\mathcal{E}(t))=0$ for $t\in \mathbb{Z}$ on the double embedding of a smooth quadric surface ${\mathbb{P}}^1\times {\mathbb{P}}^1,$ and prove the existence of them. Geometric properties of the sets of points corresponding to $\mathcal{E}$ by Serre correspondence are obtained.

Keywords:

• Cohomology
• vector bundle

2020 Mathematics Subject Classification:

• Primary: 14J60

Acknowledgments

I am very grateful to my supervisor Professor Hajime Kaji for his helpful advice and continued encouragement. I wish to thank Professor Yasunari Nagai for his many valuable comments. I would like to thank the referee for carefully reading the manuscript, correcting numerous grammatical and mathematical errors, advising me on the structure of the paper, and greatly clarifying and improving its presentation. Finally, I would also like to thank my parents for financial and emotional support.

Disclosure statement

No potential conflict of interest was reported by the author(s).