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Mikhail Zaidenberg

Publications and source records attributed to Mikhail Zaidenberg.

At least 19 recordsLinked to original sources
arXiv2026

Algebraic ellipticity of double covers of $(\mathbf P^1)^3$

We show that every smooth complex double cover of $(\mathbf P^1)^3$ branched along a smooth divisor of tridegree $(2,2,2)$ is algebraically Gromov elliptic. These irrational Fano threefolds form Mori--Mukai family 3-1 and have anticanonical genus seven and Picard rank three. Their three coordinate projections define degree-four del Pezzo fibrations. Each threefold also admits a dominant rational map of degree two from $\mathbf P^3$.

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arXiv2026

Borel subalgebras of Lie algebras of vector fields

In [I. Arzhantsev and M. Zaidenberg, Borel subgroups of the automorphism groups of affine toric surfaces, arXiv:2507.09679 (2025)] we described the Borel subgroups and maximal solvable subgroups of the automorphism groups of affine toric surfaces. In the present paper, we introduce the notion of an integrable Borel subalgebra in the Lie algebra of the automorphism group of an affine variety. We show that they are precisely the tangent algebras of the Borel subgroups. We classify the integrable Borel subalgebras in the Lie algebras of the automorphism groups of toric affine surfaces, notably of the affine plane and its cyclic quotients.

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arXiv2026

Solvability and nilpotence of ind-groups via tangent curves

Answering a question of Kraft and the second-named author, we prove that a connected ind-group over an algebraically closed field of characteristic zero is solvable, respectively nilpotent, if and only if its Lie algebra is. Their derived lengths, respectively nilpotency classes, coincide. We also obtain bounds on the derived length and nilpotency class for regular linear representations of connected ind-groups and for subgroups generated by connected algebraic subgroups. As an application, we give a new proof of the geometric case of a theorem of Belabbas: a solvable Lie algebra of vector fields on an affine variety generated by finitely many locally finite Lie subalgebras is locally finite and finite dimensional.

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arXiv2026

Algebraic ellipticity of quartic double spaces

We prove that every smooth quartic double $n$-fold, $n\ge2$, over an algebraically closed field of characteristic zero is algebraically elliptic in Gromov's sense. In particular, every smooth quartic double solid is algebraically elliptic, which answers Question~4.28 in [M. Zaidenberg, Algebraic Gromov ellipticity: a brief survey, Taiwanese J. Math. 29 (2025), no. 6, 1681-1705]. We also prove that removing any closed subset of codimension at least two preserves algebraic ellipticity for smooth quartic double spaces and smooth cubic hypersurfaces of dimension at least two.

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arXiv2026

Locally finite solvable Lie algebras of derivations

Let X be an affine variety. The local finiteness of a Lie subalgebra h of Lie(Aut(X)) is equivalent to the existence of an algebraic subgroup G of Aut(X) such that h is contained in Lie(G). Let h be a solvable Lie subalgebra of Lie(Aut(X)) generated by a finite collection of locally finite Lie subalgebras. The authors of [arXiv:2507.09679] wondered whether h is itself locally finite. After presenting some criteria for the local finiteness of h, we answer this question in the affirmative in the particular case where X is the affine plane.

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arXiv2025

Borel subgroups of the automorphism groups of affine toric surfaces

In [I. Arzhantsev and M. Zaidenberg, Acyclic curves and group actions on affine toric surfaces. Affine Algebraic Geometry, 1--41. World Scientific Publishing Co. 2013] we described the automorphism groups of the cyclic quotients of the affine plane. In this article, we study the Borel subgroups and, more generally, the maximal solvable subgroups of these ind-groups. We show that the cyclic quotients of the affine plane are divided into two species. In one of them, the Borel subgroups form a single conjugacy class, while in the other, there are two conjugacy classes of Borel subgroups. The proofs explore the Bass-Serre-Tits theory of groups acting on trees.

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arXiv2025

Automorphism groups of rigid affine surfaces: the identity component

It is known that the identity component of the automorphism group of a projective algebraic variety is an algebraic group. This is not true in general for quasi-projective varieties. In this note we address the question: given an affine algebraic surface $Y$, as to when the identity component ${\rm Aut}^0 (Y)$ of the automorphism group ${\rm Aut} (Y)$ is an algebraic group? We show that this happens if and only if $Y$ admits no effective action of the additive group of the field. In the latter case, ${\rm Aut}^0 (Y)$ is an algebraic torus of rank $\le 2$.

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arXiv2025

Algebraic Gromov's ellipticity of cubic hypersurfaces

We show that every smooth cubic hypersurface X in P^{n+1}, n> 1 is algebraically elliptic in Gromov's sense. This gives the first examples of non-rational projective manifolds elliptic in Gromov's sense. We also deduce that the punctured affine cone over X is elliptic.

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arXiv2023

Algebraically generated groups and their lie algebras

The automorphism group Aut(X) of an affine variety X is an ind-group. Its Lie algebra is canonically embedded into the Lie algebra VF(X) of vector fields on X. We study the relations between subgroups of Aut(X) and Lie subalgebras of VF(X). We show that a subgroup G of Aut(X) generated by a family of connected algebraic subgroups G_i of Aut(X) is algebraic if and only if the Lie algebras Lie G_i generate a finite dimensional Lie subalgebra of VF(X). Extending a result by Cohen-Draisma we prove that a locally finite Lie algebra L of VF(X) generated by locally nilpotent vector fields is algebraic, i.e. L = Lie G for an algebraic subgroup G of Aut(X). Along the same lines we prove that if a subgroup G of Aut(X) generated by finitely many connected algebraic groups is solvable, then it is a solvable algebraic group. We also show that the derived length a unipotent algebraic subgroup U of Aut(X) is bounded above by dim X. This result is based on the following triangulation theorem: Every unipotent algebraic subgroup of Aut(A^n) with a dense orbit in A^n is conjugate to a subgroup of the de Jonquières subgroup. Furthermore, we give an example of a free subgroup F of Aut(A^2) generated by two algebraic elements such that the Zariski closure of F is a free product of two nested commutative closed unipotent ind-subgroups. To any affine ind-group G one can associate a canonical ideal L_G \subset Lie G. It is linearly generated by the tangent spaces T_e X for all algebraic subsets X \subset G which are smooth in e. It has the important property that for a surjective homomorphism ϕ: G \to H the induced homomorphism dϕ_e : L_G \to L_H is surjective as well. Moreover, if H \subset G is a subnormal closed ind-subgroup of finite codimension, then L_H has finite codimension in L_G.

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arXiv2023

Central Weyl involutions on Fano-Mukai fourfolds of genus 10

It is known that every Fano-Mukai fourfold X of genus 10 is acted upon by an involution $τ$ which comes from the center of the Weyl group of the simple algebraic group of type ${\rm G}_2$. This involution is uniquely defined up to conjugation in the group Aut(X). In this note we describe the set of fixed points of $τ$ and the surface scroll swept out by the $τ$-invariant lines.

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arXiv2023

Gromov ellipticity of cones over projective manifolds

We find classes of projective manifolds that are elliptic in the sense of Gromov and such that the affine cones over these manifolds also are elliptic off their vertices. For example, the latter holds for any generalized flag manifold of dimension at least 3 successively blown up in a finite set of points and infinitesimally near points. This also holds for any smooth projective rational surface. For the affine cones, the Gromov ellipticity is a much weaker property than the flexibility. Nonetheless, it still implies the infinite transitivity of the action of the endomorphism monoid on these cones.

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arXiv2022

Tits-type alternative for certain groups acting on algebraic surfaces

A theorem of Cantat and Urech says that an analog of the classical Tits alternative holds for the group of birational automorphisms of a compact complex Kaehler surface. We established in our previous paper the following Tits-type alternative: if X is a toric affine variety and G is a subgroup of Aut(X) generated by a finite set of unipotent subgroups normalized by the acting torus then either G contains a nonabelian free subgroup or G is a unipotent affine algebraic group. In the present paper we extend the latter result to any group G of automorphisms of a complex affine surface generated by a finite collection of unipotent algebraic subgroups. It occurs that either G contains a nonabelian free subgroup or G is a metabelian unipotent algebraic group.

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arXiv2022

Fano-Mukai fourfolds of genus $10$ and their automorphism groups

The automorphism groups of the Fano-Mukai fourfold of genus 10 were studied in our previous paper [arXiv:1706.04926]. In particular, we found in [arXiv:1706.04926] the neutral components of these groups. In the present paper we finish the description of the discrete parts. Up to isomorphism, there are two special Fano--Mukai fourfold of genus 10 with the automorphism groups $GL_2(k)\rtimes\mathbb{Z}/2\mathbb{Z}$ and $(\mathbb{G}_a\times\mathbb{G}m)\rtimes\mathbb{Z}/2\mathbb{Z}$, respectively. For any other Fano-Mukai fourfold $V$ of genus 10 one has $\mathrm{Aut}(V)=\mathbb{G}_m^2\rtimes \mathbb{Z}/2\mathbb{Z}$, except for exactly one of them with $\mathrm{Aut}(V)=\mathbb{G}_m^2\rtimes \mathbb{Z}/6 \mathbb{Z}$.

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