Families of affine planes: the existence of a cylinder. (English) Zbl 1071.14522
From the introduction: I. V. Dolgachev and B. Yu. Weisfeiler [Izv. Akad. Nauk SSSR Ser. Mat. 38, 757–799 (1974; Zbl 0314.14015)] formulated the following
Conjecture. Let \(f:X\to S\) be a flat affine morphism of smooth schemes with every fiber isomorphic (over the residue field) to an affine space. Then \(f\) is locally trivial in the Zariski topology.
In the characteristic-0 case, this conjecture is known to be true (under much weaker assumptions) for morphisms of relative dimension 1. The known partial positive results in higher relative dimensions (see e.g., A. Sathaye [Invent. Math. 74, 159–168 (1983; Zbl 0538.13006)] and T. Asanuma and S. M. Bhatwadekar [J. Pure Appl. Algebra 115, 1–13 (1997; Zbl 0893.13006)]) deal only with families over a 1-dimensional base with 2-dimensional fibers, under an extra assumption that the generic fiber is the affine plane as well. In this paper we show that the latter assumption holds over any base. To simplify consideration, we restrict it to smooth, quasi-projective varieties defined over \(\mathbb{C}\). We say that a family \(f:X\to S\) of quasi-projective varieties contains a cylinder if, for some Zariski open subset \(S_0\) of \(S\), there is a commutative diagram \[ \begin{matrix} f^{-1}(S_0) & \overset\varphi \longrightarrow & S_0\times\mathbb{C}^k\\ f \searrow && \swarrow \text{pr}_1\\ & S_0\end{matrix} \] where \(\varphi\) is an isomorphism. Our main result is the following
Theorem 1. A smooth family \(f:X\to S\) with general fibers isomorphic to \(\mathbb{C}^2\) contains a cylinder \(S_0\times \mathbb{C}^2\).
We do not know if the theorem remains true in higher relative dimensions. A theorem of Sathaye [loc. cit.], together with theorem 1, proves the following.
Corollary. The Dolgachev-Weisfeiler conjecture is indeed true for families of affine planes over smooth curves.
On the hand, theorem 1 provides one of the principal ingredients in the proof of the following statement.
Theorem [S. Kaliman, Pac. J. Math. 203, No. 1, 161–190 (2002; Zbl 1060.14085)]. A polynomial \(p\) on \(\mathbb{C}^3\) with general fibers isomorphic to \(\mathbb{C}^2\) is a variable of the polynomial algebra \(\mathbb{C}^{[3]}\) (that is, \(\mathbb{C}^{[3]}\simeq \mathbb{C}[p]^{[2]})\). In particular, all its fibers are isomorphic to \(\mathbb{C}^2\).
Conjecture. Let \(f:X\to S\) be a flat affine morphism of smooth schemes with every fiber isomorphic (over the residue field) to an affine space. Then \(f\) is locally trivial in the Zariski topology.
In the characteristic-0 case, this conjecture is known to be true (under much weaker assumptions) for morphisms of relative dimension 1. The known partial positive results in higher relative dimensions (see e.g., A. Sathaye [Invent. Math. 74, 159–168 (1983; Zbl 0538.13006)] and T. Asanuma and S. M. Bhatwadekar [J. Pure Appl. Algebra 115, 1–13 (1997; Zbl 0893.13006)]) deal only with families over a 1-dimensional base with 2-dimensional fibers, under an extra assumption that the generic fiber is the affine plane as well. In this paper we show that the latter assumption holds over any base. To simplify consideration, we restrict it to smooth, quasi-projective varieties defined over \(\mathbb{C}\). We say that a family \(f:X\to S\) of quasi-projective varieties contains a cylinder if, for some Zariski open subset \(S_0\) of \(S\), there is a commutative diagram \[ \begin{matrix} f^{-1}(S_0) & \overset\varphi \longrightarrow & S_0\times\mathbb{C}^k\\ f \searrow && \swarrow \text{pr}_1\\ & S_0\end{matrix} \] where \(\varphi\) is an isomorphism. Our main result is the following
Theorem 1. A smooth family \(f:X\to S\) with general fibers isomorphic to \(\mathbb{C}^2\) contains a cylinder \(S_0\times \mathbb{C}^2\).
We do not know if the theorem remains true in higher relative dimensions. A theorem of Sathaye [loc. cit.], together with theorem 1, proves the following.
Corollary. The Dolgachev-Weisfeiler conjecture is indeed true for families of affine planes over smooth curves.
On the hand, theorem 1 provides one of the principal ingredients in the proof of the following statement.
Theorem [S. Kaliman, Pac. J. Math. 203, No. 1, 161–190 (2002; Zbl 1060.14085)]. A polynomial \(p\) on \(\mathbb{C}^3\) with general fibers isomorphic to \(\mathbb{C}^2\) is a variable of the polynomial algebra \(\mathbb{C}^{[3]}\) (that is, \(\mathbb{C}^{[3]}\simeq \mathbb{C}[p]^{[2]})\). In particular, all its fibers are isomorphic to \(\mathbb{C}^2\).
MSC:
| 14R25 | Affine fibrations |
| 14R10 | Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) |
