Factorial affine \(G_{\alpha}\)-varieties isomorphic to hypersurfaces of Danielewski type
type: Item
instance of: scholarly article
title: Factorial affine \(G_{\alpha}\)-varieties isomorphic to hypersurfaces of Danielewski type
zbMATH Open document ID: 1502.14147
published in: Transformation Groups
publication date: 2022-11-16T00:00:00Z
review text: 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, N…
Mathematics Subject Classification ID: 14R10
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 7618143
zbMATH Keywords: cancellation conjecture
zbMATH Keywords: Danielewski surfaces
author: K. Masuda
reviewed by: A. Ya Belov
MaRDI profile type: Publication
full work available at URL: s00031-020-09631-y
OpenAlex ID: W3102582568
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DOI: 10.1007/S00031-020-09631-Y
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description: scientific article; zbMATH DE number 7618143