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Orbits of the automorphism group of a module over a principal ideal ring
Moscow University Mathematics Bulletin. 2016. Vol. 71. No. 5. P. 196-199.
Garazha A.
Orbits of the automorphism group of a finitely generated module over a principal ideal ring are described in terms of canonical representatives and by a complete system of invariants. For a primary module, a natural bijection to a sum of two Young diagrams is established for the set of orbits and the set of partitions of the Young diagram corresponding to the module, which allows us to calculate the number of orbits.
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
Vladimir L. Popov, Proceedings of the Steklov Institute of Mathematics 2016 Vol. 292 P. 209-223
For every algebraically closed field k of characteristic different from 2, we prove
the following: (1) Finite-dimensional (not necessarily associative) k-algebras of general type
of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple
of algebraically independent (over k) rational functions of the structure constants. (2) There
exists an “algebraic normal form” to ...
Added: March 29, 2016
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
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
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
Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2015. No. 1508.02860.
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: August 13, 2015
V. L. Popov, Proceedings of the Steklov Institute of Mathematics, Springer 2019 Vol. 307 P. 193-197
We prove that, for any prime integer $p\geqslant 2$, there exists an algebraic action of the two-dimensional Witt group $W_2(p)$ on an al\-geb\-raic variety $X$ and a point $x\in X$ such that the closure of the $W_2(p)$-orbit of $x$
in $X$ contains infinitely many $W_2(p)$-orbits.\;This is related to the problem of extending from characteristic zero to ...
Added: March 30, 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
Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2017. No. 1707.06914 [math.AG].
We classify all connected affine algebraic groups G such that there are only finitely many G-orbits in every algebraic G-variety containing a dense open G-orbit. We also prove that G enjoys this property if and only if every irreducible algebraic G-variety X is modality-regular, i.e., the modality of X (in the sense of V. Arnol’d) ...
Added: July 24, 2017
Popov V. L., Mathematical notes 2019 Vol. 105 No. 3-4 P. 580-581
It is shown that the main result of N. R. Wallach, Principal orbit type theorems for reductive algebraic group actions and the Kempf–Ness Theorem, arXiv:1811.07195v1 (17 Nov 2018), is a special case of a more general statement, which can be deduced, using a short argument, from the classical Richardson and Luna theorems. ...
Added: May 27, 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
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
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., 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
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
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., 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
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
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
Vladimir L. Popov, 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
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