The divisor class group of surfaces of embedding dimension 3
type: Item
instance of: scholarly article
title: The divisor class group of surfaces of embedding dimension 3
zbMATH Open document ID: 0671.13010
DOI: 10.1016/0021-8693(88)90081-6
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let k be a field, x, y, z, u, v indeterminates, \(S=kx,y,z\), a, b units in S, and \(p,q,r,s,t\in S\). The author proves the following results: (1) If \(f=x^ 2+ay^ 3+bz^ 5+pxy^ 2+qxz^ 3+rxyz+sy^ 2z^ 2+tyz^ 4,\) then \(S/(f)\) is factorial; (2) If \(f=x^ 2+ ay^ 3+ bz^ 4+ pxy^ 2+ qxz^ 3+rxyz+sy^ 2z^ 2+tyz^ 3\) even and \(\sqrt{-b_ 1}\not\in k\), where \(b_ 1\) is the constant term of \(b\in S\), then \(S/(f)\) is factorial; (3) If \(f=x^ 2+ay^ 3+bz^ 3+pxy^ 2+qxz^ 2+rxyz+sy^ 2z^ 2\) and \(^ 3\sqrt{b_ 1/a_ 1}\not\in k\), where \(b_ 1\) is the constant term of b, \(a_ 1\) is the constant term of a, then S/(f) is factorial; (4) If \(f=x^ 2+ay^ 2+bz^ m+pxy^ 2+ qxz^{m/2+1}+ ryz^{m/2+1}+ sxyz,\) \(m\geq 2\), and \(\sqrt{-a_ 1}\), \(\sqrt{-b_ 1}\), \(\sqrt{-b_ 1/a_ 1}\not\in k\), moreover, if \(2| m\) then assume \(k^ 2\) does not contain a solution for the equation \(X^ 2+a_ 1Y^ 2=-b_ 1\). Then \(S/(f)\) is factorial. Using the results above, the author obtainsthe following corollary: \({\mathbb{R}}x,y,z/(x^ 2+ y^ 3+z^ 4)\), \({\mathbb{Z}}/3{\mathbb{Z}}x,y,z/(x^ 2+ y^ 3+z^ 4)\), \(k(u)x,y,z/(x^ 2+ y^ 2+uz^ 4)\), \({\mathbb{R}}x,y,z/(x^ 2+ y^ 2+z^ m)\), \({\mathbb{Z}}/3{\mathbb{Z}}x,y,z/(x^ 2+ y^ 2+z^{2m-1})\), \(k(u,v)x,y,z/(x^ 2+ uy^ 2+vz^ m)\), \(m\geq 1\), are factorial. Let \((R,M)\) be a 2-dimensional normal local domain which is a homomorphic image of a 3-dimensional regular local ring \((S,N)\). The author proves that the divisor class group \(Cl(R)\) of R is generated by the classes \([\bar P]\) of height 1 primes \(\bar P\) which are not contained in \(M^{e(R)}\), where \(e(R)\) is the multiplicity of R.
reviewed by: Tong Wenting
Mathematics Subject Classification ID: 13F15
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 14J20
zbMATH DE Number: 4097523
zbMATH Keywords: factorial quotient ring
zbMATH Keywords: 3-dimensional regular local ring
zbMATH Keywords: divisor class group
author: Sang-Ki Choi
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90081-6
OpenAlex ID: W2037765965
cites work: Anelli con unica decomponibilita in fattori primi ed un problema di intersezioni complete
cites work: Q5560333
cites work: Q4065643
cites work: Rational singularities, with applications to algebraic surfaces and unique factorization
cites work: Q4063557
cites work: The topology of normal singularities of an algebraic surface and a criterion for simplicity
cites work: On unique factorization domains
cites work: Über Primfaktorzerlegung in zweidimensionalen lokalen Ringen
cites work: Einige Beispiele faktorieller lokaler Ringe
cites work: Über die Divisorenklassengruppen lokaler Ringe auf zweidimensionalen normalen Hyperflächen
description: scientific article; zbMATH DE number 4097523