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Affine Space Fibrations

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Book cover Polynomial Rings and Affine Algebraic Geometry (PRAAG 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 319))

Abstract

We discuss various aspects of affine space fibrations \(f : X \rightarrow Y\) including the generic fiber, singular fibers and the case with a unipotent group action on X. The generic fiber \(X_\eta \) is a form of \({\mathbb A}^n\) defined over the function field k(Y) of the base variety. Singular fibers in the case where X is a smooth (or normal) surface or a smooth threefold have been studied, but we do not know what they look like even in the case where X is a singular surface. The propagation of properties of a given smooth fiber to nearby fibers will be studied in the equivariant case of Abhyankar-Sathaye Conjecture in dimension three. We also treat the triviality of a form of \({\mathbb A}^n\) if it has a unipotent group action. Treated subjects are classified into the following four themes

  1. 1.

    Singular fibers of \({\mathbb A}^1\)- and \({\mathbb P}^1\)-fibrations,

  2. 2.

    Equivariant Abhyankar-Sathaye Conjecture in dimension three,

  3. 3.

    Forms of \({\mathbb A}^3\) with unipotent group actions,

  4. 4.

    Cancellation problem in dimension three.

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Notes

  1. 1.

    The original conjecture asserts that if  \(f : X \rightarrow S\)  is a flat affine morphism of smooth schemes with every fiber isomorphic (over the residue field) to an affine space  \({\mathbb A}^n\),  then  f  is locally trivial in the Zariski topology. Taking credit of the conjecture, we call variants of the conjecture the Dolgachev-Weisfeiler problem.

  2. 2.

    In fact, the answer is almost complete if \(n=1\) (see [29, 30]). However, the answer remains partial if \(n=2\). A key result is Sathaye’s theorem [43], and a most general result is found in [27].

  3. 3.

    The latter is a theorem of Daigle-Kaliman [5].

  4. 4.

    Let M be the maximal ideal of \(B:=\varGamma (Y,{\mathcal O}_Y)\) corresponding to the point P. Then \(\widehat{X}\) is the blow-up of X with respect to MA, i.e., \(\widehat{X} \cong \mathrm{Proj}\,_X(\oplus _{n\ge 0}(MA)^n)\). Let \(\tau : \widehat{X} \rightarrow X\) be the canonical morphism. Then \(\tau ^{-1}(C_j)\cong C_j\times E\) for every one-dimensional component \(C_j\) of F. But \(\tau ^{-1}(S_i)\cong S_i\) for every two-dimensional component \(S_i\) of F. We note here that \(S_i\cap C_j=\emptyset \) for all i and j.

  5. 5.

    More precisely, we take a finite Galois extension \(k'/k\) such that \(D\otimes _kk'\) splits into a sum of geometrically irreducible components and consider the Galois group \(\mathrm{Gal}(k'/k)\).

  6. 6.

    The commutativity of a two-dimensional unipotent algebraic group G in the case of characteristic zero (and \(k=\overline{k}\)) follows from the triviality of the adjoint representation of G on the Lie algebra L(G) (see [20] for the general facts). By V. Popov, this is not the case in positive characteristic p since \(G=\left\{ \left( \begin{array}{ccc}1 &{} a &{} b \\ 0 &{} 1 &{}a^p \\ 0 &{} 0 &{} 1 \end{array}\right) ~ \bigg \vert ~ a, b \in k\right\} \) is a two-dimensional non-commutative unipotent group.

  7. 7.

    This means by definition that \(\overline{X}\) is factorial.

  8. 8.

    This implies that the \(G_1\)-orbit \(G_1P\) and \(G_2\)-orbit \(G_2P\) are tangent at P.

  9. 9.

    For a \(G_a\)-action \(\sigma : G_a\times X \rightarrow X\), let \(\varPsi _X:=(\sigma , p_2) : G_a\times X \rightarrow X\times X\) be the graph morphism of \(\sigma \). We say that the action \(\sigma \) is free (resp. proper) if the graph morphism \(\varPsi _X\) is a closed immersion (resp. a proper morphism). Then \(\sigma \) is free (resp. proper) if and only if \(\overline{\sigma }:=\sigma \otimes _k\overline{k}\) is free (resp. proper). By an argument as in subsection 4.4, we have the implications: \(\sigma \) is free \(\Rightarrow \) \(\sigma \) is proper \(\Rightarrow \) \(\sigma \) is fixed-point free. If X is a k-form of \({\mathbb A}^3\), a theorem of Kaliman [22] implies that these three conditions are equivalent. Hence we can use this terminology without confusion.

  10. 10.

    This follows easily from the fact that \(\overline{q} : \overline{X} \rightarrow \overline{Y}\) is an \({\mathbb A}^1\)-bundle. But we would like to make a detour to give an argument which can be applied in case we drop the hypothesis that the action \(\sigma \) is free.

  11. 11.

    The same argument in the proof of Lemma 2.6 works for \(G\cong G_a\times G_a\).

  12. 12.

    The algebraic quotient morphism \(q : X \rightarrow Y=X/\!/G\) splits as a composite \(X {\mathop {\longrightarrow }\limits ^{{q_1}}} Y_1:=X/\!/G_1 {\mathop {\longrightarrow }\limits ^{{q_2}}} Y\). If the G-action is q-tight, \(\overline{X}\) is a G-torsor over \(U:=\overline{q}(\overline{X})\) by Lemma 29. This implies that \(\overline{q}_1 : \overline{X} \rightarrow \overline{Y}_1\) is surjective and a \(G_1\)-torsor. Hence \(\overline{Y}_1\) is the geometric quotient of \(\overline{X}\) by \(G_1\). In particular, the \(G_1\)-action on X is q-tight. Since \(\dim A=4\) and \(A_1=\mathrm{Ker}\,\varDelta \) with an lnd \(\varDelta \) on A associated to the \(G_1\)-action, we need to show that \(A_1\) is finitely generated over k. By Seshadri [47], X is a locally trivial G-principal fiber bundle. Hence there exists an open covering \({\mathcal U}=\{U_i\}_{i\in I}\) with a finite index set I such that \(q^{-1}(U_i) \cong U_i\times G\) which is a G-equivalent isomorphism over \(U_i\). We can take the open sets \(U_i\) of the form \(D(b_i)\) with \(b_i \in B\). Then \(A_1[b_i^{-1}]\cong A[b_i^{-1}]^{G_1} \cong B[b_i^{-1}][t_i]\) since \(G_1\) acts on G and \(G/G_1 \cong G_a\). Hence \(A_1[b_i^{-1}]\) is generated by a single element \(f_i\) over \(B[b_i^{-1}]\). Since the positive powers of \(b_i\) generate the unitary ideal in B, it is then easy to show that \(A_1\) is generated by \(\{f_i\}_{i\in I}\) over B.

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Acknowledgements

The present article is based on the materials prepared for a research to be conducted as a Rip program 2018 in the period August 5 to September 1, 2018 at the Mathematisches Forschungsinstitute Oberwolfach (MFO). We are very much grateful to MFO for the hospitality and the nice research environment. The first author would like to thank the Department of Atomic Energy of the Government of India for Dr. Raja Ramanna Fellowship during this work. The second author is supported by Grant-in-Aid for Scientific Research (C), No. 15K04831, JSPS, and the third author is supported by Grant-in-Aid for Scientific Research (C), No. 16K05115, JSPS.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, Maharashtra, India

    Rajendra V. Gurjar

  2. Department of Mathematical Sciences, School of Science & Technology, 2-1 Gakuen, Sanda, 669-1337, Japan

    Kayo Masuda

  3. Research Center for Mathematics and Data Science, Kwansei Gakuin University, 2-1 Gakuen, Sanda, 669-1337, Japan

    Masayoshi Miyanishi

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  1. Rajendra V. Gurjar
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Correspondence to Rajendra V. Gurjar .

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Editors and Affiliations

  1. Department of Mathematical Sciences, Tokyo Metropolitan University, Hachioji, Tokyo, Japan

    Shigeru Kuroda

  2. Faculty of Applied Physics, University of Fukui, Fukui, Japan

    Nobuharu Onoda

  3. Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA

    Gene Freudenburg

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Gurjar, R.V., Masuda, K., Miyanishi, M. (2020). Affine Space Fibrations. In: Kuroda, S., Onoda, N., Freudenburg, G. (eds) Polynomial Rings and Affine Algebraic Geometry. PRAAG 2018. Springer Proceedings in Mathematics & Statistics, vol 319. Springer, Cham. https://doi.org/10.1007/978-3-030-42136-6_6

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