On the determinant of a dual periodic singular fiber

Cheng Gong; Jun Lu; Sheng-Li Tan

doi:10.4134/BKMS.b220710

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

QR
Code

Article

HOME

ALL ARTICLES

View

Bull. Korean Math. Soc. 2023; 60(5): 1365-1374

Online first article September 5, 2023 Printed September 30, 2023

https://doi.org/10.4134/BKMS.b220710

On the determinant of a dual periodic singular fiber

Cheng Gong, Jun Lu, Sheng-Li Tan

Soochow University; East China Normal University; East China Normal University

Abstract

Go to

Abstract

Let $F$ be a periodic singular fiber of genus $g$ with dual fiber $F^*$, and let $T$ (resp.~$T^*$) be the set of the components of $F$ (resp.~$F^*$) by removing one component with multiplicity one. We give a formula to compute the determinant $|\det T\,|$ of the intersect form of $T$. As a consequence, we prove that $|\det T\,|=|\det T^*\,|$. As an application, we compute the Mordell-Weil group of a fibration $f:S\to \mathbb P^1$ of genus $2$ with two singular fibers.

Keywords: Modell-Weil group, singular fiber, determinant, fibrations

MSC numbers: 14D06, 14C21, 14H10

Supported by: This work is supported by the Natural Science Foundation of Jiangsu Province(BK 20211305). This work is also supported by the National Key Research and Development Program of China (Grant No. 2018AAA0101001), the National Natural Science Foundation of China, the Shanghai Science and Technology Commission Foundation (Grants No. 22DZ2229014 and No. 20511100200).