The divisor class group of surfaces of embedding dimension 3
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.
scientific article; zbMATH DE number 4097523
scientific article