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Effective bounds for semipositive sheaves and for the height of points on curves over complex function fields. (English) Zbl 0742.14020

Let \(f:X\to Y\) be a relatively minimal surjective morphism from a smooth complex surface \(X\) to a curve \(Y\) of genus \(q\), such that \(f\) is not isotrivial and the connected general fibre \(F\) of \(f\) has genus \(g\geq 2\). The authors prove the following result: for every section \(s\) of \(f\) one has the estimate \(h(s(Y))<2(2g-1)^ 2(2q-2+2a)\), where \(a\) is the number of singular fibres of \(f\) and \(h(s(Y))\) is the height of \(s(Y)\) defined by \(h(s(Y))=\deg(s^*(\omega{X/Y}))\). If moreover \(f\) is semi-stable one has the better estimate \(h(s(Y))<2(2g-1)^ 2(2q-2+a)\). Results of this kind (besides the fact that they are interesting in themselves) have relevance in connection with Manin’s proof of the Mordell conjecture over function fields.

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
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