On the determinant of a dual periodic singular fiber. (English) Zbl 1540.14020
Summary: 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.
MSC:
| 14D06 | Fibrations, degenerations in algebraic geometry |
| 14C21 | Pencils, nets, webs in algebraic geometry |
| 14H10 | Families, moduli of curves (algebraic) |
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