Abstract
Let B be an integral domain which is finitely generated over a subdomain R and let D be an R-derivation on B such that the induced derivation \(D_{\mathfrak{m}}\) on \(B \otimes _{R}R/\mathfrak{m}\) is locally nilpotent for every maximal ideal \(\mathfrak{m}\). We ask if D is locally nilpotent. Theorem 2.1 asserts that this is the case if B and R are affine domains.We next generalize the case of G a -action treated in Theorem 2.1 to the case of \(\mathbb{A}^{1}\)-fibrations and consider the log deformations of affine surfaces with \(\mathbb{A}^{1}\)-fibrations. The case of \(\mathbb{A}^{1}\)-fibrations of affine type behaves nicely under log deformations, while the case of \(\mathbb{A}^{1}\)-fibrations of complete type is more involved [see Dubouloz–Kishimoto (Log-uniruled affine varieties without cylinder-like open subsets, arXiv: 1212.0521, 2012)].As a corollary, we prove the generic triviality of \(\mathbb{A}^{2}\)-fibration over a curve and generalize this result to the case of affine pseudo-planes of ML0-type under a suitable monodromy condition.
2000 Mathematics Subject Classification: Primary: 14R20; Secondary: 14R25
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Notes
- 1.
The result is also remarked in [3, Remark 13].
- 2.
When we write t ∈ T, we tacitly assume that t is a closed point of T.
- 3.
In order to avoid the misreading, it is better to specify our definition of simple normal crossings in the case of dimension three. We assume that every irreducible component S i of S and every fiber \(\overline{Y }_{t}\) are smooth and that analytic-locally at every intersection point P of \(S_{i} \cap S_{j}\) (resp. \(S_{i} \cap S_{j} \cap S_{k}\) or \(S_{i} \cap \overline{Y }_{t}\)), S i and S j (resp. \(S_{i},S_{j}\) and S k , or S i and \(\overline{Y }_{t}\)) behave like coordinate hypersurfaces. Hence \(S_{i} \cap S_{j}\) or \(S_{i} \cap \overline{Y }_{t}\) are smooth curves at the point P.
- 4.
We note here that without the condition on the absence of the monodromy of the cross-section, the assertion fails to hold. See Example 3.6.
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Acknowledgements
The second and third authors are supported by Grant-in-Aid for Scientific Research (C), No. 22540059 and (B), No. 24340006, JSPS.
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Gurjar, R.V., Masuda, K., Miyanishi, M. (2014). Deformations of \(\mathbb{A}^{1}\)-Fibrations. In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_19
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