Rational points of bounded height on Del Pezzo surfaces of degree six
type: Item
instance of: scholarly article
title: Rational points of bounded height on Del Pezzo surfaces of degree six
zbMATH Open document ID: 0858.14009
DOI: 10.1007/BF02566015
author: Marcello Robbiani
published in: Commentarii Mathematici Helvetici
publication date: 1996-01-28T00:00:00Z
full work available at URL: https://eudml.org/doc/140376
review text: Let \(V_3\) be a Del Pezzo surface of degree 6 over a number field \(K\). If \(W_3\) is the complement of the union of all exceptional lines of \(V_3\), denote by \(N_{W_3} (-\omega,x)\) the number of \(K\)-rational points of \(W_3\) whose anticanonical height is bounded by \(x\). The author proves the following result. Assume that all the exceptional curves of the first kind of \(V_3\) are defined over \(K\). Then for each number field \(K\) there exists a constant \(c_K\) such that \(N_{W_3} (-\omega,x) \leq c_K x(\log(x))^{3+2r}\), where \(r\) is the rank of the group of units of the ring \({\mathfrak O}_K\) of \(K\). This result is related to a conjecture of Manin.
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 838578
zbMATH Keywords: number of rational points
zbMATH Keywords: Del Pezzo surface
zbMATH Keywords: anticanonical height
zbMATH Keywords: exceptional curves
reviewed by: Lucian Bădescu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 838578