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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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