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Dyson’s lemma for products of two curves of arbitrary genus. (English) Zbl 0666.10024
Let \(C_ 1\), \(C_ 2\) be smooth irreducible complex curves of genus \(g_ 1\), \(g_ 2\), respectively. Let \({\mathcal L}_ 1\), \({\mathcal L}_ 2\) be invertible sheaves on \(C_ 1\), \(C_ 2\) respectively of degrees \(d_ 1\), \(d_ 2\); \(d_ 1\geq d_ 2\). Let \(\pi_ i: C_ 1\times C_ 2\to C_ i\) be the projection morphisms and let s be a non-identically vanishing section of \(\pi^*_ i{\mathcal L}_ 1\otimes \pi^*_ 2{\mathcal L}_ 2\). Let \(\xi_ 1,...,\xi_ m\) be points on \(C_ 1\times C_ 2\) with \(\pi_ i(\xi_ j)\neq \pi_ i(\xi_ k)\); \(i=1,2\), \(1\leq j<k\leq m\). Then, using the index and volume notation of the classical Dyson lemma, we have \[ \sum^{m}_{h=1}Vol_ h\quad \leq \quad 1+\frac{d_ 2}{2d_ 1}\max (2g_ 1-2+m,0). \] The proof uses a modification of Viola’s approach. [Cf. C. Viola, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 105–135 (1985; Zbl 0596.10032)].

MSC:
11J99 Diophantine approximation, transcendental number theory
11J68 Approximation to algebraic numbers
14H99 Curves in algebraic geometry
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References:
[1] [D] Dyson, F. J.: The approximation to algebraic numbers by rationals. Acta Math.79, 225-240 (1947) · Zbl 0030.02101 · doi:10.1007/BF02404697
[2] [E-V] Esnault, H., Viehweg, E.: Dyson’s lemma for polynomials in several variables (and the theorem of Roth). Invent. Math.78, 445-490 (1984) · Zbl 0545.10021 · doi:10.1007/BF01388445
[3] [F-W] Faltings, G., Wüstholz, G., et al.: Rational points, Seminar Bonn/Wuppertal 1983/84 (Aspects of Mathematics E6). Braunschweig: Vieweg 1984
[4] [Vi] Viola, C.: On Dyson’s lemma. Ann. Sc. Norm. Super. Pisa12, 105-135 (1985) · Zbl 0596.10032
[5] [Vo] Vojta, P.: Mordell’s conjecture over function fields. Invent. Math.98, 115-138 (1989) · Zbl 0662.14019 · doi:10.1007/BF01388847
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