Zero cycles on del Pezzo surfaces over local fields
type: Item
instance of: scholarly article
title: Zero cycles on del Pezzo surfaces over local fields
zbMATH Open document ID: 0592.14034
DOI: 10.1016/0021-8693(85)90057-2
author: Kevin R. Coombes
author: Douglas J. Muder
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let \(A_ 0(X)\) be the degree zero zero-cycles in the Chow ring of X. By studying \(H^ 1(Pic\)(X\({}_ E))\) the author proves: Theorem 1. Let X be a del Pezzo surface of degree \(d\geq 5\), defined over a local field k. Then \(A_ 0(X)=0.\) Theorem 2. A fine moduli space for the set of marked split nonsingular del Pezzo surfaces is \(M={\mathbb{P}}^ 2-C\), where C is the union of lines through four fixed points. To a point (a;b;c)\(\in M\) we associate the surface \(bsv=ar(u-t)+(e-b)su\), \(btv=cu(r-s)+(a-b)rt.\) Theorem 3. Let E/k be an unramified quadratic extension of local fields. Let X be a degree 4 del Pezzo k-surface split by E. Then either \(A_ 0(X)=0\) or X is a type IV surface with bad reduction.
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14C22
zbMATH DE Number: 3950721
zbMATH Keywords: Picard group
zbMATH Keywords: zero-cycles in the Chow ring
zbMATH Keywords: Pic
zbMATH Keywords: del Pezzo surface
zbMATH Keywords: quadratic extension of local fields
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90057-2
OpenAlex ID: W2066147151
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description: scientific article; zbMATH DE number 3950721