### ?

## Automorphism groups of affine varieties consisting of algebraic elements

Proceedings of the American Mathematical Society. 2024. Vol. 152. No. 6. P. 2377–2383.

Perepechko A., Regeta A.

Given an affine algebraic variety X, we prove that if the neutral component Aut◦(X) of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group G contains a closed connected nested ind-subgroup H ⊂ G, and for any g ∈ G some positive power of g belongs to H, then G = H.

Perepechko A., Regeta A., Transformation Groups 2023 Vol. 28 P. 401–412

For an affine algebraic variety X, we study the subgroup Autalg(X) of the group of regular automorphisms Aut(X) of X generated by all the connected algebraic subgroups. We prove that Autalg(X) is nested, i.e., is a direct limit of algebraic subgroups of Aut(X), if and only if all the Ga-actions on X commute. Moreover, we ...

Added: October 28, 2022

Perepechko A., Michałek M., Süß H., Mathematische Zeitschrift 2018 Vol. 290 No. 3-4 P. 1457–1478

We provide a new criterion for flexibility of affine cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre–Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces. ...

Added: September 26, 2019

Ignatev Mikhail, Penkov I., Transformation Groups 2022

We compute the group of automorphisms of an arbitrary ind-variety of (possibly
isotropic) generalized flags. Such an ind-variety is a homogeneous ind-space for
one of the ind-groups SL(∞), O(∞) or Sp(∞). We show that the respective automorphism groups are much larger than SL(∞), O(∞) or Sp(∞), and present the
answer in terms of Mackey groups. The latter are ...

Added: October 8, 2023

Shramov K., Prokhorov Y., / Cornell University. Series arXiv "math". 2019.

We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite subgroups. We also prove that the stabilizer of a point in the automorphism group of a compact complex surface of zero Kodaira dimension, as well as the stabilizer of a point in the automorphism group of an arbitrary compact Kaehler manifold of ...

Added: November 19, 2019

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2013. No. 1307.5522.

This is an expanded version of my talk at the workshop ``Groups of Automorphisms in Birational and Affine Geometry'', October 29–November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. ...

Added: July 21, 2013

Popov V. L., Zarhin Y., / Cornell University. Series math "arxiv.org". 2018. No. 1808.01136.

We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\mathcal L(K)$ generated by ${\rm Aut} (K)$ and multipli\-ca\-tions by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are ...

Added: August 8, 2018

Nina I. Zhukova, Anna Yu. Dolgonosova .., Central European Journal of Mathematics 2013 Vol. 11 No. 12 P. 2076–2088

The category of foliations is considered. In this category
morphisms are differentiable mappings transforming leaves of one
foliation into leaves of the other foliation.
We proved that the automorphism group of the foliations
admitting a transverse linear connection is an infinite-dimensional
Lie group modeled on $LF$-spaces. This result extends the corresponding
result of Macias-Virgos and Sanmartin for Riemannian foliations.
In particular, our ...

Added: September 28, 2014

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2014. No. 1401.0278.

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...

Added: January 3, 2014

Kuyumzhiyan K., Proceedings of the American Mathematical Society 2020 No. 148 P. 3723–3731

We prove the conjecture of Berest-Eshmatov-Eshmatov by showing that the group of automorphisms of a product of Calogero-Moser spaces C_n_i, where the n_i are pairwise distinct, acts m-transitively for each m. ...

Added: August 18, 2020

Avilov A., Математические заметки 2020 Т. 107 № 1 С. 3–10

The forms of the Segre cubic over non-algebraically closed fields, their automorphisms groups, and equivariant birational rigidity are studied. In particular, it is shown that all forms of the Segre cubic over any field have a point and are cubic hypersurfaces. ...

Added: May 11, 2020

Arzhantsev I., Perepechko A., / Bulletin des sciences mathématiques. Series 22-00305 "BULSCI-D". 2023.

We consider complete toric varieties X such that a maximal unipotent subgroup U of the automorphism group Aut(X) acts on X with an open orbit. It turns out that such varieties can be characterized by several remarkable properties. We study the set of Demazure roots of the corresponding complete fan, describe the structure of a maximal unipotent subgroup U in Aut(X), and find all regular subgroups ...

Added: October 6, 2023

Nikolay Konovalov, / Cornell University. Series "Working papers by Cornell University". 2022. No. 2202.07507.

Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence ...

Added: September 12, 2022

Arzhantsev I., Zaitseva Y., Research in Mathematical Sciences 2024 Vol. 11 No. 2 Article 27

An algebraic variety X is called a homogeneous variety if the automorphism group Aut(X) acts on X transitively, and a homogeneous space if there exists a transitive action of an algebraic group on X. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we ...

Added: March 23, 2024

Vladimir L. Popov, Transformation Groups 2014 Vol. 19 No. 2 P. 549–568

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...

Added: March 17, 2014

Prokhorov Y., Cheltsov I., / Cornell University. Series arXiv "math". 2020.

We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups. ...

Added: August 19, 2020

Vladimir L. Popov, Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 185–213

This is an expanded version of my talk at the workshop
``Groups of Automorphisms in Birational and Affine Geometry'',
October 29–November 3, 2012, Levico Terme, Italy.
The first section is focused on Jordan groups in abstract setting,
the second on that in the settings of automorphisms groups and
groups of birational self-maps of algebraic varieties.
The appendix is an expanded version ...

Added: April 28, 2014

Arzhantsev I., Communications in Algebra 2008 Vol. 36 No. 12 P. 4368–4374

Added: July 10, 2014

Perepechko A., Математические заметки 2021 Т. 110 № 5 С. 744–750

Affine algebraic surfaces of Markov type of the form
x^2 + y^2 + z^2 − xyz = c
are studied. Their automorphism groups are found. ...

Added: October 12, 2021

Shramov K., Przyjalkowski V., Proceedings of the Steklov Institute of Mathematics 2019 Vol. 307 P. 198–209

We show that smooth well-formed weighted complete intersections have finite automorphism groups, with several obvious exceptions. ...

Added: August 12, 2020

Arzhantsev I., Ricerche di Matematica 2024 Vol. 73 No. 2 P. 715–724

We show that an effective action of the one-dimensional torus G_m on a normal affine algebraic variety X can be extended to an effective action of a semi-direct product G_m⋌G_a with the same general orbit closures if and only if there is a divisor D on X that consists of G_m-fixed points. This result is applied to the study of orbits of the automorphism group Aut(X) on X. ...

Added: August 16, 2021

Avilov A., Sbornik Mathematics 2016 Vol. 307 No. 3 P. 315–330

We prove that any G-del Pezzo threefold of degree 4, except for a one-parameter family and four distinguished cases, can be equivariantly reconstructed to the projective space ℙ3, a quadric Q ⊂ ℙ4 , a G-conic bundle or a del Pezzo fibration. We also show that one of these four distinguished varieties is birationally rigid ...

Added: July 6, 2016

Trushin A., Математические заметки 2023 Т. 113 № 5 С. 780–784

It is known that the group of automorphisms of the algebra of polynomials in three variables is not generated by elementary automorphisms. In the article, a system of generators is constructed for the group of automorphisms preserving the nontrivial grading by integers. ...

Added: September 18, 2024

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

Gayfullin S., Шафаревич А. А., / Cornell University. Series arXiv "math". 2018. No. arXiv:1805.05024.

Added: September 1, 2018