The arithmetic of zero cycles on surfaces with geometric genus and irregularity zero
type: Item
instance of: scholarly article
title: The arithmetic of zero cycles on surfaces with geometric genus and irregularity zero
zbMATH Open document ID: 0787.14002
DOI: 10.1007/BF01445218
author: Kevin R. Coombes
published in: Mathematische Annalen
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/164877
review text: Let \(X\) be a smooth, projective, geometrically irreducible surface over a perfect field \(F\). Throughout this paper, it will be assumed that the geometric genus \(p_ g\) and the irregularity \(q\) of \(X\) both vanish. Denote the separable closure of \(F\) by \(\overline{F}\). Let \(\overline{X} = X \otimes_ F\overline{F}\) be the surface obtained from \(X\) by base extension. It will also be assumed that the group \(A_ 0(\overline X)\) of rational equivalence classes of zero cycles of degree zero on \(\overline{X}\) vanishes. This is a technical hypothesis which could presumably be eliminated at the expense of working with \(\text{Ker}(A_ 0(X)\to A_ 0(\overline{X}))\). For want of a better name, and for ease in stating various results, any surface which satisfies these three hypotheses will be called a pseudo-rational surface. -- \textit{S. Bloch} [Commun. Algebra 3, 405-428 (1975; Zbl 0327.14002)] has conjectured that the vanishing of \(A_ 0(\overline{X})\) should follow from the assumption that \(p_ g = q = 0\). This was proven by \textit{S. Bloch, A. Kas} and \textit{D. Lieberman} [Compos. Math. 3, 135-145 (1976; Zbl 0337.14006)] for all such surfaces which are not of general type. The class of pseudo-rational surfaces includes: rational surfaces, Enriques surfaces, elliptic surfaces with \(q=0\), the classical Godeaux surface, Burniat-Inoue surfaces, Campedelli surfaces, and the surfaces of Barlow and of Keum. This paper will study \(A_ 0(X)\) for pseudo-rational surfaces defined over fields of number theoretic interest. One of the main results of this paper is a new proof of the following theorem of \textit{W. Raskind} [in Algebraic \(K\)-theory: Connections with geometry and topology, Proc. Meet., Lake Louise/Can. 1987, NATO ASI Ser., Ser. C 279, 343-388 (1989; Zbl 0709.14005)]. Theorem 0.1. Let \(X\) be a pseudo-rational surface over a non-archimedean local field \(L\) with residue field \(F\). Assume that \(X\) has very good reduction to a smooth project…
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 59964
zbMATH Keywords: geometric genus
zbMATH Keywords: irregularity
zbMATH Keywords: rational equivalence classes of zero cycles
zbMATH Keywords: pseudo-rational surface
zbMATH Keywords: very good reduction
Wikidata QID: Q57556832
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 59964