Some remarks on algebraic equivalence of cycles
type: Item
instance of: scholarly article
title: Some remarks on algebraic equivalence of cycles
zbMATH Open document ID: 0542.14002
author: Alberto Collino
author: Giuseppe Ceresa
published in: Pacific Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: Let \(F\subset {\mathbb{P}}_ 4({\mathbb{C}})\) be a threefold with exactly one singular point, which is an ordinary double point p, and F' be the proper transform of F under the blowing up of \({\mathbb{P}}_ 4({\mathbb{C}})\) at p: F' is smooth, and the inverse image of p in F' is a smooth quadric H with two lines L and M belonging to the two distinct rulings of H. This paper shows that if F is general, of degree at least five, then L and M are not algebraically equivalent in F', although they are homologically equivalent. A result of the same type is obtained for the general non- singular quintic threefolds in \({\mathbb{P}}_ 4({\mathbb{C}})\).
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14K30
zbMATH DE Number: 3861307
zbMATH Keywords: homological equivalence
zbMATH Keywords: algebraic equivalence
zbMATH Keywords: threefold with exactly one singular point
zbMATH Keywords: quintic threefolds
reviewed by: Frédéric Campana
MaRDI profile type: Publication
full work available at URL: pjm.1983.105.285
OpenAlex ID: W2027742713
DOI: 10.2140/PJM.1983.105.285
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description: scientific article; zbMATH DE number 3861307