\(R\)-equivalence on low degree complete intersections
type: Item
instance of: scholarly article
zbMATH Open document ID: 1253.14047
DOI: 10.1090/S1056-3911-2011-00581-X
author: Alena Pirutka
published in: Journal of Algebraic Geometry
publication date: 2012-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/0907.1971
Mathematics Subject Classification ID: 14M22
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 6086797
zbMATH Keywords: Rationally connected varieties
zbMATH Keywords: Chow groups
zbMATH Keywords: \(R\)-equivalence
MaRDI profile type: Publication
arXiv ID: 0907.1971
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OpenAlex ID: W2135962722
title: \(R\)-equivalence on low degree complete intersections
review text: Let \(k\) be a field of characteristic zero. Let \(X\) be a projective geometrically integral variety over \(k\) with a fixed ample divisor, and let \(\bar{M}_{0,2}(X,d)\) denote the Kontsevich moduli space for all genus zero stable curves over \(X\) of degree \(d\) with two marked points. We say that \(X\) is \(k\)-rationally simply connected if \(H_2(X,\mathbb{Z})\) has rank one and for any sufficiently large integer \(e\) there exists a geometrically irreducible component \(M_{e,2}\subset\bar{M}_{0,2}(X,e)\) intersecting the open locus of irreducible curves \(M_{0,2}(X,e)\) and such that the restriction of the evaluation morphism \(M_{e,2}\rightarrow X\times X\) is dominant with rationally connected general fiber. In this paper the author proves that if \(k\) is either a function field in one variable over \(\mathbb{C}\) or the field \(\mathbb{C}((t))\) and \(X\) is a \(k\)-rationally simply connected variety over \(k\), then \(R\)-equivalence on rational points of \(X\) is trivial and that the Chow group of zero-cycles of degree zero is trivial too. In particular, this holds for a smooth complete intersection of \(r\) hypersurfaces in \(\mathbb{P}_k^n\) of respective degrees \(d_1,\dots,d_r\) with \(\sum_{i=1}^rd_i^2\leq n+1\).
reviewed by: Fumio Hazama
description: scientific article; zbMATH DE number 6086797