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Extension of maps defined on many fibres. (English) Zbl 0937.14003

From the introduction: Let \(S\) be a surface and let \(\pi:S\to B\) be a fibration of curves of genus \(g\leq 2\). Assume that for non countably many \(t\in B\), the fibre \(F_t\) is endowed with a non-constant morphism \(\varphi_t :F_t\to D_t\) into a smooth curve. The goal of this note is to show that the existence of these maps \(\varphi_t\) implies the existence of another fibration \(T \to B\) and of a rational map over \(B\) from \(S\) to \(T\) reflecting the properties of many \(\varphi_t\). In fact we recover the original maps \(\varphi_t\) only for non countably many values of \(t\) (even in the case that one applies this under the hypothesis of existence of \(\varphi_t\) for a general \(t\), one cannot get better results as simple examples show). To obtain the surface \(T\) we shall need base change in general. However under some hypothesis of unicity this base change can be avoided.
We consider three cases: First we assume that the maps \(\varphi_t\) are automorphisms (theorem 1.2); secondly we suppose that \(1 \leq g(D_t)<g\) (theorem 2.4), and finally we study linear series (theorem 3.1). We also obtain a similar result for abelian schemes with abelian subvarieties in the general fibres (theorem 2.5).

MSC:

14D06 Fibrations, degenerations in algebraic geometry
14J10 Families, moduli, classification: algebraic theory
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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