×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] H. Esnault and E. Viehweg , Dyson’s lemma for polynomials in several variables (and the theorem of Roth) . Invent. Math. 78 (1984) 445-490. · Zbl 0545.10021
[2] H. Esnault and E. Viehweg , Logarithmic De Rham complexes and vanishing theorems . Invent. Math. 86 (1986) 161-194. · Zbl 0603.32006
[3] T. Fujita , On Kähler fibre spaces over curves . J. Math. Soc. Japan 30 (1978) 779-794. · Zbl 0393.14006
[4] H. Grauert , Mordell’s Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper . Publ. Math. IHES 25 (1965) 131-149. · Zbl 0137.40503
[5] Y. Kawamata , A generalization of Kodaira-Ramanujam’s vanishing theorem . Math. Ann. 261 (1982) 57-71. · Zbl 0476.14007
[6] Yu. I. Manin , Rational points on an algebraic curve over function fields . Trans. Amer. Math. Soc. 50 (1966) 189-234. · Zbl 0178.55102
[7] S. Mori , Classification of higher-dimensional varieties. Algebraic Geometry . Bowdoin 1985. Proc. of Symp. in Pure Math. 46 (1987) 269-331. · Zbl 0656.14022
[8] A.N. Parshin , Algebraic curves over function fields I. Math. USSR Izv. 2 (1968) 1145-1170. · Zbl 0188.53003
[9] L. Szpiro , Séminaire sur les pinceaux de courbes de genre au moins deux . Astérisque 86 (1981). · Zbl 0463.00009
[10] E. Viehweg , Vanishing theorems . J. Reine Angew. Math. 335 (1982) 1-8. · Zbl 0485.32019
[11] E. Viehweg , Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces . Adv. Stud. Pure Math. 1 (1983) 329-353 North-Holland. · Zbl 0513.14019
[12] E. Viehweg , Vanishing theorems and positivity in algebraic fibre spaces . Proc. Intern. Congr. Math., Berkeley 1986, 682-687. · Zbl 0685.14013
[13] E. Viehweg , Weak positivity and the stability of certain Hilbert points . Invent. Math. 96 (1989) 639-667. · Zbl 0695.14006
[14] P. Vojta , Mordell’s conjecture over function fields . Preprint 1988. · Zbl 0662.14019
[15] A.N. Parshin , Algebraic curves over function fields . Soviet Math. Dokl. 9 (1968) 1419-1422. · Zbl 0176.50903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.