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Zbl 0742.14020
Esnault, Hélène; Viehweg, Eckart
Effective bounds for semipositive sheaves and for the height of points on curves over complex function fields. (English)
[J] Compos. Math. 76, No.1-2, 69-85 (1990). ISSN 0010-437X; ISSN 1570-5846/e

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 $g\ge 2$. The authors prove the following result: for every section $s$ of $f$ one has the estimate $h(s(Y))<2(2g-1)\sp 2(2q-2+2a)$, where $a$ is the number of singular fibres of $f$ and $h(s(Y))$ is the height of $s(Y)$ defined by $h(s(Y))=\deg(s\sp*(\omega{X/Y}))$. If moreover $f$ is semi-stable one has the better estimate $h(s(Y))<2(2g-1)\sp 2(2q-2+a)$. 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.
[L.Bădescu (Bucureşti)]

MSC 2000:: *14G40 Arithmetic varieties and schemes
14F05 Sheaves, etc.
14J20 Arithmetic ground fields (surfaces)

Keywords: geometric heights; section of surjective morphisms; Mordell conjecture over function fields

Cited in: Zbl 0772.14009

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