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Algebraic groups and the Cremona group
P. 1053–1055.
The following topics about subgroups of the Cremona groups are discussed: (1) maximal tori; (2) conjugacy and classification of diagonalizable subgroups of codimensions 0 and 1; (3) conjugacy of finite abelian subgroups; (4) algebraicity of normalizers of diagonalizable subgroups; (5) torsion primes.
Language:
English
In book
Vol. 10. Issue 2. , European Mathematical Society Publishing house, 2013.
Ivan Arzhantsev, Roman Avdeev, Yulia Zaitseva, International Mathematics Research Notices 2026 Vol. 2026 No. 4 Article rnag007
We complete the classification of algebraic monoid structures on the affine 3-space. The result is based on a reduction of the general case to that of commutative monoids. We also study various algebraic properties of all monoids appearing in the classification. ...
Added: February 24, 2026
Popov V., Успехи математических наук 2025 Т. 80 № 3(483) С. 189–190
It is proved that for any positive integers d and c, the set of isomorphism classes of all d-dimensional reductive algebraic groups with exactly c connected components is finite. As a corollary, the set of isomorphism classes of all d-dimensional compact real Lie groups with exactly c connected components is proved to be finite. To ...
Added: December 16, 2025
Arzhantsev I., Izvestiya. Mathematics 2009 Vol. 73 No. 3 P. 437–453
We study open equivariant projective embeddings of homogeneous spaces such that the complement of the open orbit has codimension at least 2. We establish a criterion for the existence of such an embedding, prove that the set of isomorphism classes of such embeddings is finite, and give a construction of the embeddings in terms of ...
Added: June 13, 2025
Arzhantsev I., Sbornik Mathematics 2001 Vol. 192 No. 8 P. 1133–1138
Let G be a reductive algebraic group and let H be a reductive subgroup of G. The modality of a G-variety X is the largest number of the parameters in a continuous family of G-orbits in X. A precise formula for the maximum value of the modality over all affine embeddings of the homogeneous space ...
Added: June 13, 2025
Zaitseva Y., Results in Mathematics 2024 Vol. 79 Article 249
We give a classification of noncommutative algebraic monoid structures on normal affine varieties such that the group of invertible elements of the monoid is connected, solvable, and has a one-dimensional unipotent radical. We describe the set of idempotents and the center of such a monoid and give a criterion for existence of the zero element. ...
Added: September 13, 2024
Roman Avdeev, Communications in Contemporary Mathematics 2024 Vol. 26 No. 6 Article 2350029
In this paper, we obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain one-parameter degenerations of the Lie algebras corresponding to such subgroups. As an application, we exhibit explicit algorithms for computing the set of spherical ...
Added: December 27, 2023
Gayfullin S., Chunaev D., Фундаментальная и прикладная математика 2023 Т. 24 № 4 С. 47–59
In this work we obtain sufficient conditions for a variety with a torus action of complexity one to have finite number of automorphism group orbits. ...
Added: December 2, 2023
Beldiev I., Results in Mathematics 2023 Vol. 78 No. 5 Article 192
We study induced additive actions on projective hypersurfaces, i.e. effective regular actions of the algebraic group G_a^m with an open orbit that can be extended to a regular action on the ambient projective space. We prove that if a projective hypersurface admits an induced additive action, then it is unique if and only if the hypersurface is ...
Added: August 7, 2023
Cham: Springer, 2023.
This book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and Pohang
The conferences were focused on the following two related problems:
• existence of Kähler–Einstein metrics on Fano varieties
• degenerations of Fano varieties
on which two famous conjectures were recently proved. The first is the famous ...
Added: May 24, 2023
Vladimir L. Popov, / Series math "arxiv.org". 2023. No. 2302.13374.
We prove that the Picard group of a connected affine algebraic group $G$ is isomorphic to the fundamental group of the derived subgroup of the reductive algebraic group $G/{\mathscr R}_u(G)$, where ${\mathscr R}_u(G)$ is the unipotent radical of $G$. ...
Added: February 28, 2023
Arzhantsev I., Zaitseva Y., Russian Mathematical Surveys 2022 Vol. 77 No. 4 P. 571–650
We survey recent results on open embeddings of the affine space C^n into a complete algebraic variety X such that the action of the vector group G_a^n on C^n by translations extends to an action of G_a^n on X. We begin with the Hassett–Tschinkel correspondence describing equivariant embeddings of \mathbb{C}^n into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag ...
Added: February 26, 2023
Ivan V. Arzhantsev, Yulia I. Zaitseva, Kirill V. Shakhmatov, Proceedings of the Steklov Institute of Mathematics 2022 Vol. 318 No. 1 P. 13–25
Let X be an algebraic variety such that the group Aut(X) acts on X transitively. We define the transitivity degree of X as the maximum number m such that the action of Aut(X) on X is m-transitive. If the action of Aut(X) is m-transitive for all m, the transitivity degree is infinite. We compute the transitivity degree for all quasi-affine toric varieties and for many homogeneous spaces of algebraic groups. We also discuss a conjecture and ...
Added: November 5, 2022
Arzhantsev I., Zaitseva Y., Shakhmatov K., Труды Математического института им. В.А. Стеклова РАН 2022 Т. 318 С. 17–30
Let X be an algebraic variety such that the group Aut(X) acts on X transitively. We define the transitivity degree of X as a maximal number m such that the action of Aut(X) on X is m-transitive. If the action of Aut(X) is m-transitive for all m, the transitivity degree is infinite. We compute the transitivity degree for all quasi-affine toric varieties and for many homogeneous spaces of algebraic groups. Also we discuss a ...
Added: November 4, 2022
Vladimir L. Popov, / Series math "arxiv.org". 2022. No. 2207.13072.
A new infinite series of rational affine algebraic varieties is constructed whose automorphism group contains the auto\-morphism group Aut(F_n) of the free group F_n of rank n. The automorphism groups of such varieties are nonlinear and contain the braid group B_n on n strands for n > 2, and are nonamenable for n > 1. ...
Added: July 27, 2022
Vladimir L. Popov, / Series math "arxiv.org". 2022. No. 2206.14040.
We prove that every orbit of the adjoint representation of any connected reductive algebraic group G is a rational algebraic variety. For complex simply connected semisimple G, this implies rationality of affine Hamiltonian G-varieties (which we classify). ...
Added: June 29, 2022
Popov V., Известия РАН. Серия математическая 2022 Т. 86 № 5 С. 73–96
We explore to what extent the group variety of a connected algebraic group or the group manifold of a real Lie group determines its group structure. ...
Added: June 9, 2022
Arzhantsev I., Shakhmatov K., Results in Mathematics 2022 Vol. 77 No. 2 Article 75
In this note we prove that every finite collection of connected algebraic subgroups of the group of triangular automorphisms of the affine space generates a connected solvable algebraic subgroup. ...
Added: February 16, 2022
V. L. Popov, Doklady Mathematics, USA 2021 Vol. 104 No. 2 P. 264–266
We explore to what extent the group variety of an algebraic group determines its group structure. ...
Added: December 24, 2021
В. Л. Попов, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2021 Т. 500 № 1 С. 52–54
It is explored to which extent the group variety of an algebraic group determines its group structure. ...
Added: November 18, 2021
Roman Avdeev, Transformation Groups 2021 Vol. 26 No. 2 P. 403–431
Given a connected reductive complex algebraic group G and a spherical subgroup H⊂G, the extended weight monoid Λˆ+G(G/H) encodes the G-module structures on spaces of regular sections of all G-linearized line bundles on G/H. Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P⊂G, ...
Added: September 9, 2021
Vladimir L. Popov, / Series math "arxiv.org". 2021. No. 2105.12861.
Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. ...
Added: May 28, 2021
Vladimir L. Popov, / Series math "arxiv.org". 2021. No. 2102.08032.
Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained. ...
Added: February 17, 2021