Dyson's lemma for products of two curves of arbitrary genus
type: Item
instance of: scholarly article
title: Dyson's lemma for products of two curves of arbitrary genus
zbMATH Open document ID: 0666.10024
DOI: 10.1007/BF01388846
author: Paul Vojta
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143722
review text: 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 \(d_1\), \(d_2\); \(d_1\ge d_2\). 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 \(\xi_1,\ldots,\xi_m\) be points on \(C_1\times C_2\) with \(\pi_i(\xi_j)\ne \pi_i(\xi_k)\); \(i=1,2\), \(1\le j<k\le m\). Then, using the index and volume notation of the classical Dyson lemma, we have \[ \sum^{m}_{h=1} \text{Vol}_ h\quad \le 1+\frac{d_2}{2d_1}\max (2g_1-2+m,0). \] The proof uses a modification of Viola's approach. [Cf. \textit{C. Viola}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 105--135 (1985; Zbl 0596.10032)].
reviewed by: Paul Vojta
Mathematics Subject Classification ID: 11J97
Mathematics Subject Classification ID: 11J68
Mathematics Subject Classification ID: 14H99
zbMATH DE Number: 4089651
zbMATH Keywords: smooth irreducible complex curves
zbMATH Keywords: Dyson lemma
MaRDI profile type: Publication
Wikidata QID: Q124814004
cites work: The approximation to algebraic numbers by rationals
cites work: Dyson's lemma for polynomials in several variables (and the theorem of Roth)
cites work: Q3728083
cites work: Mordell's conjecture over function fields
OpenAlex ID: W1999461552
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description: scientific article; zbMATH DE number 4089651