Affine homogeneous varieties and suspensions. (English) Zbl 07824473
The paper is devoted to the study of discrepancy between algebraic homogeneous spaces and algebraic varieties with transitive action of the automorphism group. For complete varieties these two classes coincide. However, in general, the second class is strictly bigger: all smooth quasi-affine toric varieties are acted on transitively by the automorphism group, but some of them are not homogeneous spaces under an algebraic group action [I. V. Arzhantsev et al., Sb. Math. 203, No. 7, 923–949 (2012; Zbl 1311.14059); translation from Mat. Sb. 203, No. 7, 3–30 (2012)]. In a previous paper [I. V. Arzhantsev et al., Proc. Steklov Inst. Math. 318, 13–25 (2022; Zbl 1506.14121); translation from Tr. Mat. Inst. Steklova 318, 17–30 (2022)], the authors raise a question whether such examples exist for affine varieties.
Here they give an affirmative answer to this question based on the construction of suspension: an affine variety \(X\) is said to be a suspension over another affine variety \(Y\) if \(X\) is defined in \(\mathbb{A}^2\times Y\) by an equation of the form \(uv=f(y)\), where \(u,v\) are the coordinates on \(\mathbb{A}^2\) and \(f\) is a non-constant regular function on \(Y\). The suspension \(X\) is smooth if and only if \(Y\) and the zero scheme of \(f\) on \(Y\) are smooth.
There is a notion of flexibility for affine varieties [I. Arzhantsev et al., Duke Math. J. 162, No. 4, 767–823 (2013; Zbl 1295.14057)]: one of equivalent definitions is that the automorphism group is transitive on the smooth locus. It is known that flexibility is inherited by suspensions [I. V. Arzhantsev et al., Sb. Math. 203, No. 7, 923–949 (2012; Zbl 1311.14059); translation from Mat. Sb. 203, No. 7, 3–30 (2012)]. Thus smooth iterated suspensions over flexible varieties are acted on transitively by the automorphism group. Taking \(Y=\mathbb{A}^{n-1}\) and a polynomial \(f\) whose prime factors are pairwise distinct and define \(m\) pairwise disjoint smooth hypersurfaces in \(\mathbb{A}^{n-1}\), one of them being a hyperplane, the authors get a smooth \(n\)-dimensional affine variety \(X\) with \(\operatorname{Pic}(X)=\mathbb{Z}^{m-1}\) such that \(\operatorname{Aut}(X)\) acts transitively on \(X\). They prove that \(\operatorname{rk}\operatorname{Pic}(X)\le\dim X\) for any affine homogeneous space \(X\), thus getting plenty of examples of affine varieties with transitive automorphism group which are not homogeneous spaces.
In particular, those (and only those) Danielewski surfaces which are suspensions over \(\mathbb{A}^1\) have transitive automorphism group. It is proved that among them only a smooth affine quadric is a homogeneous space. It is also proved that Danilov-Gizatullin surfaces which are obtained by removing an ample curve with self-intersection index \(\ge3\) from a Hirzebruch surface are not homogeneous spaces, still having the transitive automorphism group [M. H. Gizatullin, Math. USSR, Izv. 4, 787–810 (1971; Zbl 0219.14023)].
Here they give an affirmative answer to this question based on the construction of suspension: an affine variety \(X\) is said to be a suspension over another affine variety \(Y\) if \(X\) is defined in \(\mathbb{A}^2\times Y\) by an equation of the form \(uv=f(y)\), where \(u,v\) are the coordinates on \(\mathbb{A}^2\) and \(f\) is a non-constant regular function on \(Y\). The suspension \(X\) is smooth if and only if \(Y\) and the zero scheme of \(f\) on \(Y\) are smooth.
There is a notion of flexibility for affine varieties [I. Arzhantsev et al., Duke Math. J. 162, No. 4, 767–823 (2013; Zbl 1295.14057)]: one of equivalent definitions is that the automorphism group is transitive on the smooth locus. It is known that flexibility is inherited by suspensions [I. V. Arzhantsev et al., Sb. Math. 203, No. 7, 923–949 (2012; Zbl 1311.14059); translation from Mat. Sb. 203, No. 7, 3–30 (2012)]. Thus smooth iterated suspensions over flexible varieties are acted on transitively by the automorphism group. Taking \(Y=\mathbb{A}^{n-1}\) and a polynomial \(f\) whose prime factors are pairwise distinct and define \(m\) pairwise disjoint smooth hypersurfaces in \(\mathbb{A}^{n-1}\), one of them being a hyperplane, the authors get a smooth \(n\)-dimensional affine variety \(X\) with \(\operatorname{Pic}(X)=\mathbb{Z}^{m-1}\) such that \(\operatorname{Aut}(X)\) acts transitively on \(X\). They prove that \(\operatorname{rk}\operatorname{Pic}(X)\le\dim X\) for any affine homogeneous space \(X\), thus getting plenty of examples of affine varieties with transitive automorphism group which are not homogeneous spaces.
In particular, those (and only those) Danielewski surfaces which are suspensions over \(\mathbb{A}^1\) have transitive automorphism group. It is proved that among them only a smooth affine quadric is a homogeneous space. It is also proved that Danilov-Gizatullin surfaces which are obtained by removing an ample curve with self-intersection index \(\ge3\) from a Hirzebruch surface are not homogeneous spaces, still having the transitive automorphism group [M. H. Gizatullin, Math. USSR, Izv. 4, 787–810 (1971; Zbl 0219.14023)].
Reviewer: Dmitri A. Timashev (Moskva)
MSC:
| 14M17 | Homogeneous spaces and generalizations |
| 14R20 | Group actions on affine varieties |
| 14J50 | Automorphisms of surfaces and higher-dimensional varieties |
| 14L30 | Group actions on varieties or schemes (quotients) |
Keywords:
affine algebraic variety; homogeneous space; automorphism group; transitivity; suspension; Danielewski surface; Danilov-Gizatullin surface; Picard groupReferences:
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