Abstract
A Gizatullin surface is a normal affine surface V over ℂ, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of ℂ* -actions and double-struck A1 -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with ℂ+ -actions on V considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one double-struck A 1 -fibration V → S up to an isomorphism of the base S. Moreover, an effective ℂ* -action on them, if it does exist, is unique up to conjugation and inversion t → -1 of ℂ*. Obviously, uniqueness of ℂ* -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of ℂ* -actions and double-struck A1 -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when double-struck A1 -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base S ≅ double-struck A 1. We exhibit as well large subclasses of Gizatullin ℂ* -surfaces for which a ℂ* -action is essentially unique and for which there are at most two conjugacy classes of double-struck A1 -fibrations over double-struck A1.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 305-354 |
| Number of pages | 50 |
| Journal | Transformation Groups |
| Volume | 13 |
| Issue number | 2 |
| DOIs | |
| State | Published - Jun 2008 |
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ASJC Scopus subject areas
- Mathematics(all)
Cite this
Uniqueness of ℂ* - and ℂ+ -actions on Gizatullin surfaces. / Flenner, Hubert; Kaliman, Shulim; Zaidenberg, Mikhail.
In: Transformation Groups, Vol. 13, No. 2, 06.2008, p. 305-354.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Uniqueness of ℂ* - and ℂ+ -actions on Gizatullin surfaces
AU - Flenner, Hubert
AU - Kaliman, Shulim
AU - Zaidenberg, Mikhail
PY - 2008/6
Y1 - 2008/6
N2 - A Gizatullin surface is a normal affine surface V over ℂ, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of ℂ* -actions and double-struck A1 -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with ℂ+ -actions on V considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one double-struck A 1 -fibration V → S up to an isomorphism of the base S. Moreover, an effective ℂ* -action on them, if it does exist, is unique up to conjugation and inversion t → -1 of ℂ*. Obviously, uniqueness of ℂ* -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of ℂ* -actions and double-struck A1 -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when double-struck A1 -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base S ≅ double-struck A 1. We exhibit as well large subclasses of Gizatullin ℂ* -surfaces for which a ℂ* -action is essentially unique and for which there are at most two conjugacy classes of double-struck A1 -fibrations over double-struck A1.
AB - A Gizatullin surface is a normal affine surface V over ℂ, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of ℂ* -actions and double-struck A1 -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with ℂ+ -actions on V considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one double-struck A 1 -fibration V → S up to an isomorphism of the base S. Moreover, an effective ℂ* -action on them, if it does exist, is unique up to conjugation and inversion t → -1 of ℂ*. Obviously, uniqueness of ℂ* -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of ℂ* -actions and double-struck A1 -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when double-struck A1 -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base S ≅ double-struck A 1. We exhibit as well large subclasses of Gizatullin ℂ* -surfaces for which a ℂ* -action is essentially unique and for which there are at most two conjugacy classes of double-struck A1 -fibrations over double-struck A1.
UR - http://www.scopus.com/inward/record.url?scp=49049095886&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=49049095886&partnerID=8YFLogxK
U2 - 10.1007/s00031-008-9014-0
DO - 10.1007/s00031-008-9014-0
M3 - Article
VL - 13
SP - 305
EP - 354
JO - Transformation Groups
JF - Transformation Groups
SN - 1083-4362
IS - 2
ER -