Factorial affine \(G_{\alpha}\)-varieties isomorphic to hypersurfaces of Danielewski type
The author belongs to school of Miyanishi Masayoshi and his research is focused on calceelation conjecture type questions. Famous and one of most important in affine algebraic geometry \textit{cancellation conjecture} says that if \(\mathbb{A}^1\times X\simeq \mathbb{A}^{n+1}\) then \(X\simeq \mathbb{A}^{n}\). Danielewski fond series of varieties \(X_i, Y_i\) called \textit{Danielewski spaces} such that \(X_i\not\simeq Y_i\) but \(X_i\times\mathbb{A}^1 \simeq Y_i\times\mathbb{A}^1\). In positive characteristics counterexample for cancellation conjecture was found by \textit{N. Gupta} [Compos. Math. 150, No. 6, 979--998 (2014; Zbl 1327.14251)]. One of important technique is working with action of additive group \(\mathbb{G}_a\) or -- equivalently -- locally nilpotent derivations. This provides the paper motivation. Let \(k\) be an algebraically closed field of characteristic zero and \(B\) a factorial affine \(k\)-domain equipped with a locally nilpotent derivation \(\delta\). The author investigates \(B\) when there exists an element \(z \in B\) such that \(\delta (z) = \alpha^p\) for \(p \geq 1\) and a prime element \(\alpha\) of \(A = \mathrm{Ker}\; \delta\). Motivation is provided by an example of particular interest is the coordinate ring of an affine pseudo-\(n\)-space, which is defined as a smooth affine variety \(X\) equipped with a faithfully at morphism \(q :X \rightarrow \mathbb{A}^1\) such that \(q^{-1}(\mathbb{A}_{\ast}^1) \cong \mathbb{A}_{\ast}^1 \times \mathbb{A}^{n-1}\) and \(q^* (0)\) is irreducible and reduced where \(\mathbb{A}_{\ast}^1 = \mathbb{A}^1\setminus \{0\}\). In [\textit{R. V. Gurjar} et al., J. Pure Appl. Algebra 216, No. 2, 296--313 (2012; Zbl 1237.14073)], a criterion was given for an affine pseudo-\(3\)-space to be isomorphic to a hypersurface of form \(x^p y - g(x, t, z) = 0\) in \(\mathbb{A}^4\). Such a hypersurface is called a \textit{hypersurface of Danielewski type} and studied in [\textit{K. Masuda}, Proc. Am. Math. Soc. 145, No. 4, 1439--1452 (2017; Zbl 1390.14190); J. Pure Appl. Algebra 218, No. 4, 624--633 (2014; Zbl 1291.14089)]. In this paper, under the condition that \(A/\alpha A\) is factorial, author describes \(B\) in terms of equivariant affine modification developed by \textit{Sh. Kaliman} and \textit{M. Zaidenberg} [Transform. Groups 4, No. 1, 53--95 (1999; Zbl 0956.14041)] and give a criterion for \(B\) to be isomorphic to the residue ring \(A[Y,Z]/(\alpha^p Y-g(Z))\) for \(p \geq 1\) and an irreducible polynomial \(g(Z) \in A[Z]\setminus A\). As a consequence, he obtains a criterion for an affine pseudo-\(n\)-space to be isomorphic to a hypersurface of Danielewski type for \(n \geq 3\).
scientific article; zbMATH DE number 7618143
scientific article