Homology planes without cancellation property
\textit{Danielewski} first constructed a family of smooth affine surfaces \(\{V_ n\}\) \((n\geq 1)\) such that \(V_ n\times\mathbb{C}\) are all isomorphic, where \(V_ n\) is an affine hypersurface \(x^ nz+y^ 2=1\) in \(\mathbb{C}^ 3\), and \textit{Fieseler} showed that \(V_ n\) and \(V_ m\) are not homeomorphic if \(n\neq m\). Generalizing the examples of Danielewski in the equivariant context, the author constructs families of \(\mathbb{Q}\)-homology planes without the cancellation property. We shall explain a part of these results. Let \(a\geq 2\) and \(b\geq 1\) be integers and let \(W(a,b)\) be an affine hypersurface in \(\mathbb{C}^ 3\) defined by \(x^ bz+y^ a=1\). The cyclic group \(\mathbb{Z}/(a)\) identified with the group of the \(a\)-th roots of the unity acts on \(W(a,b)\) via \(\lambda\cdot(x,y,z)=(\lambda x,\lambda y,\lambda^{-b}z)\). Define \(V(a,b)\) as the quotient variety of \(W(a,b)\) by this action of \(\mathbb{Z}/(a)\). Then it is shown that \(V(a,b)\) is a smooth affine surface with \(\mathbb{C}\)-fibration (whence \(V(a,b)\) has Kodaira dimension \(-\infty)\), \(\pi_ 1\bigl(V(a,b)\bigr)\cong\mathbb{Z}/(a)\) and \(H_ i\bigl(V(a,b)\bigr)=0\) for \(i>1\). Thus, \(V(a,b)\) is a \(\mathbb{Q}\)- homology plane but not a homology plane. [Note that a homology plane with Kodaira dimension \(-\infty\) is \(\mathbb{C}^ 2\).] Moreover, for integers \(a\), \(a'\geq 2\) and \(b,b'\geq 1\) such that \(2/a+1/b\leq 1\) and \(2/a'+1/b'\leq 1\), \(V(a,b)\) is homeomorphic to \(V(a',b')\) if and only if \((a,b)=(a',b')\). Furthermore, \(V(a,b)\times\mathbb{C}\cong V(a,b')\times\mathbb{C}\). Thus the cancellation does not hold for \(V(a,b)\) and \(V(a,b')\) with \(b\neq b'\).
families of \(\mathbb{Q}\)-homology planes
affine hypersurface in \(\mathbb{C}^ 3\)
smooth affine surface with \(\mathbb{C}\)-fibration
scientific article; zbMATH DE number 119843