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Recent developments in affine algebraic geometry
, 2007
"... We shall review recent developments in affine algebraic geometry. The topics treated in the present article cover only a part of this vast area of research. ..."
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We shall review recent developments in affine algebraic geometry. The topics treated in the present article cover only a part of this vast area of research.
ON THE UNIQUENESS OF C∗ACTIONS ON AFFINE SURFACES
, 2004
"... It is an open question whether every normal affine surface V over C admits an effective action of a maximal torus T = C ∗n (n ≤ 2) such that any other effective C ∗action is conjugate to a subtorus of T in Aut(V). We prove that this holds indeed in the following cases: (a) the MakarLimanov invaria ..."
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It is an open question whether every normal affine surface V over C admits an effective action of a maximal torus T = C ∗n (n ≤ 2) such that any other effective C ∗action is conjugate to a subtorus of T in Aut(V). We prove that this holds indeed in the following cases: (a) the MakarLimanov invariant ML(V) ̸ = C is nontrivial, (b) V is a toric surface, (c) V = P 1 × P 1 \∆, where ∆ is the diagonal, and (d) V = P 2 \Q, where Q is a nonsingular quadric. In case (a) this generalizes a result of Bertin for smooth surfaces, whereas (b) was previously known for the case of the affine plane (Gutwirth [Gut]) and (d) is a result of DanilovGizatullin [DG] and Doebeli [Do].
JORDAN PROPERTY FOR CREMONA GROUPS
"... Abstract. Assuming Borisov–Alexeev–Borisov conjecture, we prove that there is a constant J = J(n) such that for any rationally connected variety X of dimension n and any finite subgroup G ⊂ Bir(X) there exists a normal abelian subgroup A ⊂ G of index at most J. In particular, we obtain that the Crem ..."
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Abstract. Assuming Borisov–Alexeev–Borisov conjecture, we prove that there is a constant J = J(n) such that for any rationally connected variety X of dimension n and any finite subgroup G ⊂ Bir(X) there exists a normal abelian subgroup A ⊂ G of index at most J. In particular, we obtain that the Cremona group Cr3 = Bir(P3) enjoys the Jordan property. 1.
Smooth affine surfaces with nonunique . . .
, 2008
"... In this paper we complete the classification of effective C ∗actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion λ ↦ → λ −1 of C ∗. If a smooth affine surface V admits more than one C ∗action then it is known to be Gizatullin i.e., it can be comp ..."
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In this paper we complete the classification of effective C ∗actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion λ ↦ → λ −1 of C ∗. If a smooth affine surface V admits more than one C ∗action then it is known to be Gizatullin i.e., it can be completed by a linear chain of smooth rational curves. In [FKZ3] we gave a sufficient condition, in terms of the DolgachevPinkhamDemazure (or DPD) presentation, for the uniqueness of a C ∗action on a Gizatullin surface. In the present paper we show that this condition is also necessary, at least in the smooth case. In fact, if the uniqueness fails for a smooth Gizatullin surface V which is neither toric nor DanilovGizatullin, then V admits a continuous family of pairwise nonconjugated C ∗actions depending on one or two parameters. We give an explicit description of all such surfaces and their C ∗actions in terms of DPD presentations. We also show that for every k> 0 one can find a DanilovGizatullin surface V (n) of index n = n(k) with a family of pairwise nonconjugate
Smooth affine surfaces with . . .
, 2008
"... In this paper we complete the classification of effective C ∗actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion λ ↦ → λ −1 of C ∗. If a smooth affine surface V admits more than one C ∗action then it is known to be Gizatullin i.e., it can be com ..."
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In this paper we complete the classification of effective C ∗actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion λ ↦ → λ −1 of C ∗. If a smooth affine surface V admits more than one C ∗action then it is known to be Gizatullin i.e., it can be completed by a linear chain of smooth rational curves. In [FKZ3] we gave a sufficient condition, in terms of the DolgachevPinkhamDemazure (or DPD) presentation, for the uniqueness of a C ∗action on a Gizatullin surface. In the present paper we show that this condition is also necessary, at least in the smooth case. In fact, if the uniqueness fails for a smooth Gizatullin surface V which is neither toric nor DanilovGizatullin, then V admits a continuous family of pairwise nonconjugated C ∗actions depending on one or two parameters. We give an explicit description of all such surfaces and their C ∗actions in terms of DPD presentations. We also show that for every k> 0 one can find a DanilovGizatullin surface V (n) of index n = n(k) with a family of pairwise nonconjugate
Actions of C∗ and C+ on affine algebraic varieties
 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS
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SELECTED PROBLEMS MIKHAIL ZAIDENBERG
, 2005
"... We start with the following complexanalytic version of the famous Poincaré Conjecture. It was repeatedly proposed in [Za1] and [Za2]. Problem 1. Let D be a strictly pseudoconvex bounded domain in C 2 with a smooth boundary S. Suppose S is a homology sphere. Is it true that S is diffeomorphic to S 3 ..."
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We start with the following complexanalytic version of the famous Poincaré Conjecture. It was repeatedly proposed in [Za1] and [Za2]. Problem 1. Let D be a strictly pseudoconvex bounded domain in C 2 with a smooth boundary S. Suppose S is a homology sphere. Is it true that S is diffeomorphic to S 3? One may assume in addition that S is real analytic or even real algebraic, that D is contractible, and ask whether D is a sublevel set of a real (strictly subharmonic) Morse function with just one critical point. As a motivation, we consider in loc.cit. an exhaustion of a smooth contractible affine surface X by a sequence of strictly pseudoconvex subdomains Dn ⊆ X with realalgebraic boundaries Sn. If X is not isomorphic to C 2 (for instance, if X is the Ramanujam surface) then for all n large enough, Sn is a homology sphere with a nontrivial fundamental group. Thus we expect that such a domain Dn cannot be biholomorphically equivalent to a bounded domain in C 2. The next 7 problems and conjectures also address contractible or acyclic varieties; see [Za2] for additional motivations. Problem 2. Consider a smooth contractible surface X of loggeneral type. Is the set