Abstract
The possible structure of singular fibers of an \(\mathbb{A}^{1}\)-fibration on a smooth affine surface is well understood, in particular, any such fiber is a disjoint union of affine lines (possibly with multiplicities). This paper lies in a three-dimensional generalization of this fact, i.e., properties concerning a fiber component of a given fibration f: X → B from a smooth affine algebraic threefold X onto a smooth algebraic curve B whose general fibers are affine surfaces admitting \(\mathbb{A}^{1}\)-fibrations. The phenomena differ according to the type of \(\mathbb{A}^{1}\)-fibrations on general fibers of f (namely, of affine type, or of complete type). More precisely, in case of affine type, each irreducible component of every fiber of f: X → B admits an effective \(\mathbb{G}_{a}\)-action provided Pic(X) = (0) with some additional conditions concerning a compactification, whereas for the complete type, there exists an example in which a special fiber of \(f: \mathbb{A}^{3} \rightarrow \mathbb{A}^{1}\) possesses no longer an \(\mathbb{A}^{1}\)-fibration.
MSC2010: 14R25, 14R20, 14D06, 14J30.
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Notes
- 1.
Recall that ML(S) is defined to be an intersections in \(\Gamma (\mathcal{O}_{S})\) of \(\text{Ker}(\partial )\)’s when \(\partial \) ranges over all of locally nilpotent derivations of \(\Gamma (\mathcal{O}_{S})\).
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Acknowledgements
The author would like to thank the referees, Professors Mikhail Zaidenberg and Adrien Dubouloz, for giving him useful and constructive comments consistently. Especially, in the previous version, the author has got wrong with the usage of the crucial result [11, Theorem 2.8], and the referees advised to apply the result appropriately and further persuaded him not to go in a wrong direction with an example. Without their advices and patience, this chapter can never be completed. The part of this work was done during the author’s stay at l’Institut de Mathématiques de Bourgogne (Dijon). The author would like to express his hearty thanks to the institute for giving him a splendid research environment. The author was supported by a Grant-in-Aid for Scientific Research of JSPS No. 24740003.
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Kishimoto, T. (2014). Remark on Deformations of Affine Surfaces with \(\mathbb{A}^{1}\)-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_20
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