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ON THE DETERMINANT OF A DUAL PERIODIC SINGULAR FIBER

Cheng Gong|Jun Lu|Sheng-Li Tan
대한수학회 (The Korean Mathematical Society)|2023.09|대한수학회보 (Bulletin of the KMS)|제60권 제5호|pp. 1365-1374|DOI 10.4134/BKMS.b220710

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참고문헌

키워드

determinantModell-Weil groupsingular fiberfibrations
주제분류수학

초록

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 $f:S\to \bP^1$ of genus $2$ with two singular fibers.

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 $f:S\to \bP^1$ of genus $2$ with two singular fibers.

참고문헌 (16)

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