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## Variations on the Theme of Zariski’s Cancellation Problem

P. 233-250.

This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12--16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equi-variant versions. We discuss several topics inspired by this exploration, including the problem of classifying a class of affine algebraic groups that are naturally singled out in studying the conjugacy problem for algebraic subgroups of the Cremona groups.

### In book

Vol. 319: Polynomial Rings and Affine Algebraic Geometry, PRAAG 2018, Tokyo, Japan, February 12−16. , Springer, 2020

Popov V., Doklady Mathematics 2018 Vol. 98 No. 2 P. 413-415

The compressibility of certain types of finite groups of birational automorphisms of algebraic varieties is established. ...

Added: November 13, 2018

Popov V., Journal of the Ramanujan Mathematical Society 2013 Vol. 28A No. Special Issue-2013 dedicated to C.S.Seshadri's 80th birthday P. 409-415

We construct counterexamples to the rationality conjecture regarding the new version of the Makar-Limanov invariant formulated in A. Liendo, G_a-actions of fiber type on affine T-varieties, J. Algebra 324 (2010), 3653--3665. ...

Added: June 20, 2013

Zhgoon V., Journal of Lie Theory 2013 Vol. 23 P. 607-638

Let $G$ be a connected reductive group acting on an irreducible normal algebraic variety $X$. We give a slightly improved version of Local Structure Theorems obtained by Knop and Timashev, which describe the action of some parabolic subgroup of $G$ on an open subset of $X$. We also extend various results of Vinberg and Timashev ...

Added: February 6, 2013

Popov V., Underlying varieties and group structures / Cornell University. 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

Popov V., , in: Advanced Studies in Pure Mathematics. Vol. 75: Algebraic Varieties and Automorphism Groups.: Tokyo: American Mathematical Society, World Scientific, 2017.. P. 425-441.

Exploring Bass’ Triangulability Problem on unipotent algebraic
subgroups of the affine Cremona groups, we prove a triangulability
criterion, the existence of nontriangulable connected solvable affine algebraic
subgroups of the Cremona groups, and stable triangulability
of such subgroups; in particular, in the stable range we answer Bass’
Triangulability Problem in the affirmative. To this end we prove a theorem
on invariant subfields ...

Added: July 12, 2017

Popov V., Documenta Mathematica 2015 Vol. Extra Volume: Merkurjev's Sixtieth Birthday P. 513-528

A “rational” version of the strengthened form of the Commuting Derivation Conjecture, in which the assumption of commutativity
is dropped, is proved. A systematic method of constructing in any dimension greater than 3 the examples answering in the negative a question by M. El Kahoui is developed. ...

Added: September 25, 2015

Popov V., Izvestiya: Mathematics, England 2019 Vol. 83 No. 4 P. 830-859

The first group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in
other groups. The third concerns the connectedness of the Cremona groups. ...

Added: September 29, 2019

Popov V., Pacific Journal of Mathematics 2015 Vol. 279 No. 1--2 (Special issue In memoriam: Robert Steinberg) P. 423-446

For the coordinate algebras of connected affine algebraic groups, we explore
the problem of finding a presentation by generators and relations canonically
determined by the group structure. ...

Added: December 27, 2015

Elagin A. D., Sbornik Mathematics 2012 Vol. 203 No. 5 P. 645-676

We put forward a method for constructing semiorthogonal decompositions of the derived category of G-equivariant sheaves on a variety X under the assumption that the derived category of sheaves on X admits a semiorthogonal decomposition with components preserved by the action of the group G on X. This method is used to obtain semiorthogonal decompositions ...

Added: February 4, 2013

Popov V., Embeddings of groups Aut(F_n) into automorphism groups of algebraic varieties / Cornell University. Series math "arxiv.org". 2021. No. 2106.02072.

For each integer n>0, we construct a series of irreducible algebraic varieties X, for which the automorphism group Aut(X) contains as a subgroup the automorphism group Aut(F_n) of a free group F_n of rank n. For n > 1, such groups Aut(X) are nonamenable, and for n > 2, they are nonlinear and contain the ...

Added: June 7, 2021

Popov V., 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

Arzhantsev I., Braun L., Hausen J. et al., European Journal of Mathematics 2018 Vol. 4 No. 1 P. 242-312

Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of surface singularities. We extend this picture to log terminal singularities in any dimension ...

Added: March 4, 2018

Arzhantsev I., Kotenkova P., Documenta Mathematica 2015 Vol. 20 P. 1039-1053

Added: October 19, 2015

Popov V., Birational splitting and algebraic group actions / Cornell University. Series math "arxiv.org". 2015. No. 1502.02167.

According to the classical theorem, every irreducible algebraic variety endowed with a nontrivial rational action of a connected linear algebraic group is birationally isomorphic to a product of another algebraic variety and the s-dimensional projectice space with positive s. We show that the classical proof of this theorem actually works only in characteristic 0 and ...

Added: February 10, 2015

Popov V., 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

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

Added: July 10, 2014

Amerik E., Campana F., On families of lagrangian tori on hyperkaehler manifolds / . 2013.

This is a note on Beauville's problem (solved by Greb, Lehn and Rollenske in the non-algebraic case and by Hwang and Weiss in general) whether a lagrangian torus on an irreducible holomorphic symplectic manifold is a fiber of a lagrangian fibration. We provide a different, very short solution in the non-algebraic case and make some ...

Added: April 9, 2013

Arzhantsev I., Celik D., Hausen J., Journal of Algebra 2013 Vol. 387 P. 87-98

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. ...

Added: November 13, 2013

Popov V., 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

Popov V., Известия РАН. Серия математическая 2019 Т. 84 № 4 С. 194-225

The rst group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in
other groups. The third concerns the connectedness of the Cremona groups. ...

Added: July 31, 2019

Arzhantsev I., Liendo A., Stasyuk T., Journal of Pure and Applied Algebra 2021 Vol. 225 No. 2 P. 106499

Let X be a normal variety endowed with an algebraic torus action. An additive group action alpha on X is called vertical if a general orbit of alpha is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of alpha in Aut(X). Our first result in this paper ...

Added: July 29, 2020

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

Popov V., European Journal of Mathematics 2016 Vol. 2 No. 1 P. 283-290

According to the classical theorem, every algebraic variety
endowed with a nontrivial rational action of a connected linear algebraic
group is birationally isomorphic to a product of another algebraic variety
and the projective space of a positive dimension. We show that the classical proof of this theorem
actually works only in characteristic 0 and we give a characteristic free
proof ...

Added: February 2, 2016

Popov V., Rationality and the FML invariant / Bielefeld University. Series LAGRS "Linear Algebraic Groups and Related Structures". 2012. No. 485.

We construct counterexamples to the rationality conjecture regar-ding the new version of the Makar-Limanov invariant introduced in A. Liendo, Ga-actions of fiber type on affine T-varieties, J. Algebra 324 (2010), 3653–3665. ...

Added: January 9, 2013