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## On automorphism groups of affine surfaces

P. 207-286.

This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic groups. At the same time, they can be studied from the viewpoint of the combinatorial group theory, so we put a special accent on group-theoretical aspects (ind-groups, amalgams, etc.). We provide different approaches to classification, prove certain new results, and attract attention to several open problems.

### In book

Vol. 75: Algebraic Varieties and Automorphism Groups. , Tokyo : American Mathematical Society, World Scientific, 2017

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

Perepechko A., Mathematical notes 2021 Vol. 110 No. 5 P. 732-737

Affine algebraic surfaces of Markov type of the form (Formula presented.) are studied. Their automorphism groups are found. ...

Added: October 28, 2022

Shramov K., European Journal of Mathematics 2019

We show that automorphism groups of Hopf and Kodaira surfaces have unbounded
finite subgroups. For elliptic fibrations on Hopf, Kodaira, bielliptic, and K3 surfaces,
we make some observations on finite groups acting along the fibers and on the base
of such a fibration. ...

Added: December 11, 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

Zhukova N., Moscow Mathematical Journal 2018

We introduce a category of rigid geometries on singular spaces which
are leaf spaces of foliations and are considered as leaf manifolds. We
single out a special category F_0 of leaf manifolds containing the orbifold
category as a full subcategory. Objects of F_0 may have non-Hausdorff
topology unlike the orbifolds. The topology of some objects of F_0 does
not satisfy ...

Added: April 2, 2018

Tokyo : American Mathematical Society, World Scientific, 2017

Preface
The workshop “Algebraic Varieties and Automorphism Groups” was held at the Research Institute of Mathematical Sciences (RIMS), Kyoto University during July 7-11, 2014. There were over eighty participants including twenty people from overseas Canada, France, Germany, India, Korea, Poland, Russia, Singapore, Switzerland, and USA.
Recently, there have been remarkable developments in algebraic geometry and related fields, ...

Added: July 12, 2017

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

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

Prokhorov Y., Shramov K., / Cornell University. Series arXiv "math". 2018.

We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan. ...

Added: June 8, 2019

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

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

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

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

Sheina K., / Cornell University. Series arXiv "math". 2020. No. 04348v1.

The basic automorphism group of a Cartan foliation (M, F) is the quotient group of the automorphism group of (M, F) by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism groups. Estimates ...

Added: December 9, 2020

Zhukova N., В кн. : Международная молодежная школа-семинар "Современная геометрия и ее приложения". Международная конференция "Современная геометрия и ее приложения". Материалы школы-семинара и конференции. : Каз. : Издательство Казанского университета, 2017. С. 48-51.

We introduce a category of rigid geometries on smooth singular spaces of leaves of foliations.
A special category $\mathfrak F_0$ containing orbifolds is allocated. Unlike orbifolds, objects
of $\mathfrak F_0$ can have non-Hausdorff topology and can even not satisfy the separability axiom $T_0$.
It is shown that the rigid geometry $(N,\zeta)$, where $N\in (\mathfrak F_0)$, allows a desingularization. ...

Added: April 1, 2018

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, / 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

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

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

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

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., 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

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., Zaidenberg M., European Journal of Mathematics 2022 Vol. 8 No. 2 P. 561-572

The automorphism groups of the Fano–Mukai fourfold of genus 10 were studied in our previous paper (Prokhorov and Zaidenberg in Eur J Math 4(3):1197–1263, 2018). In particular, we found the neutral components of these groups. In the present paper we finish the description of the discrete parts. Up to isomorphism, there are three special Fano–Mukai ...

Added: November 28, 2022