Affine curves on which every point is a set-theoretic complete intersection
type: Item
instance of: scholarly article
title: Affine curves on which every point is a set-theoretic complete intersection
zbMATH Open document ID: 0542.14013
DOI: 10.1016/0021-8693(84)90163-7
author: Edward D. Davis
author: Paolo Maroscia
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Given a 1-dimensional function field K and field of constants k, let A be an entire k-algebra of finite type with field of fractions K, and X the affine curve Spec A. Well known facts: Pic A is a torsion group (i.e., A is prefactorial) \(\Leftrightarrow\) each point of X is set-theoretically the complete intersection of X and a hypersurface (in one, and so every, affine embedding of X); Pic A is torsion if k is algebraic over a finite field. Two theorems for the case of algebraically closed k (the first due to Cunnea, the second to Murthy and Pedrini): \(Pic A=0\) (i.e., A is factorial) \(\Leftrightarrow each\) point of X is principal (i.e., scheme- theoretically the complete intersection of X and a hypersurface) \(\Leftrightarrow X\) is rational (i.e., \(K=k(t))\) and smooth \(\Leftrightarrow X\) is the affine line minus a finite set of points; in case char \(k=0\), Pic A is torsion \(\Rightarrow Pic A=0.\) This paper treats related results which do not necessarily restrict k as much. Among the main technical tools used are: the units-Pic Mayer-Vietoris sequence arising from the ''conductor context'' [as in the Murthy-Pedrini treatment; cf. \textit{M. P. Murthy} and \textit{C. Pedrini}, Algebr. K-Theory II, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 342, 109-121 (1973; Zbl 0283.14003)]; in the case of normal X, the exact sequence (due essentially to F. K. Schmidt) involving Pic A, the group of units of A, the group of degree 0 divisors of K mod linear equivalence, and the set of points at infinity with respect to X; and certain classical geometric methods - albeit thoroughly disguised - involving the linear series on a projective completion of X. To convey something of the flavor of the results obtained, here are an example and, for the case of algebraically closed k, three theorems which supplement the two stated above: Example. Let \(X=C-S\), where C is the complete plane cuspidal cubic over k and S is a finite set of simple points of C. Let r be the rank…
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14C22
zbMATH DE Number: 3861318
zbMATH Keywords: affine curve
zbMATH Keywords: complete intersection point
zbMATH Keywords: geometrically unibranch singular point
zbMATH Keywords: rationality
zbMATH Keywords: rank of Pic
zbMATH Keywords: units
zbMATH Keywords: conductor
zbMATH Keywords: Mayer- Vietoris sequence
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90163-7
OpenAlex ID: W1994484069
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description: scientific article; zbMATH DE number 3861318