### ?

## Rationality of three-dimensional quotients by monomial actions

Journal of Algebra. 2010. Vol. 324. No. 9. P. 2166-2197.

Kang M., Yuri Prokhorov

Vladimir L. Popov, Generic algebras: rational parametrization and normal forms / Cornell University. Series math "arxiv.org". 2014. No. 1411.6570.

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent over $\boldsymbol k$ rational functions in the structure constants. ...

Added: November 25, 2014

Trepalin A., Central European Journal of Mathematics 2014

Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary ...

Added: October 14, 2013

Iliev A., Katzarkov L., Victor Przyjalkowski, Proceedings of the Edinburgh Mathematical Society 2014 Vol. 57 P. 145-173

This paper suggests a new approach to questions of rationality of threefolds based on category theory. Following M. Ballard, D. Favero, L. Katzarkov (ArXiv:1012.0864) and D. Favero, L. Katzarkov (Noether--Lefschetz Spectra and Algebraic cycles, in preparation) we enhance constructions from A. Kuznetsov (arXiv:0904.4330) by introducing Noether--Lefschetz spectra --- an interplay between Orlov spectra (C. Oliva, ...

Added: July 2, 2013

Bogomolov F. A., Böhning C., Graf von Bothmer H., Central European Journal of Mathematics 2012 Vol. 10 No. 2 P. 466-520

Let G be one of the groups SL n(ℂ), Sp 2n(ℂ), SO m(ℂ), O m(ℂ), or G 2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G × ℙ N is rational. In this paper we improve known bounds for the levels of stable ...

Added: February 6, 2013

Nabebin A. A., Tarasikov A. S., M. : -, 2012

The manual contains necessary data from homogeneous and inhomogeneous universal algebras, systems of axioms for the basic algebraic structures (arithmetics, monoids, subgroups, groups, partial ordering, rings, fields). It is described axiomatic programming language OBJ3 with examples of programs in this language. It is intended to students of universities of specialities: the applied mathematics, informatics, computer ...

Added: March 13, 2013

Vladimir L. Popov, 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

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

Andrey S. Trepalin, Central European Journal of Mathematics 2014 Vol. 12 No. 2 P. 229-239

Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary ...

Added: December 3, 2013

Galkin S., Shinder E., The Fano variety of lines and rationality problem for a cubic hypersurface / Cornell University. Series math "arxiv.org". 2014. No. 1405.5154.

We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational ...

Added: May 21, 2014

Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al., Compositio Mathematica 2011 Vol. 147 No. 2 P. 428-466

Let k be a field of characteristic zero, let G be a connected reductive algebraic group
over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k rational functions on G, respectively, g. The conjugation action of G on itself induces
the adjoint action of G on g. We investigate the ...

Added: March 17, 2013

Vyugin I. V., Математические заметки 2019 Т. 106 № 2 С. 212-221

An upper bound for the number of field elements that can be taken to roots of unity of fixed multiplicity by means of several given polynomials is obtained. This bound generalizes the bound obtained by V'yugin and Shkredov in 2012 to the case of polynomials of degree higher than 1. This bound was obtained both over ...

Added: October 9, 2019

Макарычев С. В., Vyugin I. V., Arnold Mathematical Journal 2019 Vol. 5 No. 1 P. 105-121

We present an upper bound on the number of solutions of an algebraic equation P(x,y)=0 where x and y belong to the union of cosets of some subgroup of the multiplicative group κ∗ of some field of positive characteristic. This bound generalizes the bound of Corvaja and Zannier (J Eur Math Soc 15(5):1927–1942, 2013) to the case of union of cosets. We also ...

Added: August 13, 2019

В. Л. Попов, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2016 Т. 468 № 5 С. 499-501

A general theorem on the purity of of invariant field extensions
is proved. Using it, a criterion of rational triangulability of connected solvable
affine algebraic subgroups of the Cremona groups is obtained. This criterion is
applied for proving the existence of rationally nontriangulable subgroups of the
above form and for proving their stable rational triangulability. The latter property
answers in ...

Added: May 16, 2016

Kuznetsov A., Perry A., Compositio Mathematica 2018 Vol. 154 No. 7 P. 1362-1406

We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, ...

Added: September 13, 2018

Loginov K., Математические заметки 2019 Т. 106 № 6 С. 881-893

We consider threefold del Pezzo fibrations over a curve germ whose central fiber is non-rational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change and show that there is a 1-to-1 correpspondence between such fibrations and certain non-singular del Pezzo fibrations ...

Added: October 29, 2019

Prokhorov Y., Rationality of Fano threefolds with terminal Gorenstein singularities, I / Cornell University. Series arXiv "math". 2019.

We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found. ...

Added: November 19, 2019

Kuznetsov A., Prokhorov Y., Rationality of Fano threefolds over non-closed fields / Cornell University. Series arXiv "math". 2019.

We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic. ...

Added: August 19, 2020

V. L. Popov, Zarhin Y. G., Doklady Mathematics 2018 Vol. 98 No. 3 P. 600-602

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 generated by Aut(K) and multiplications by the elements of K*. ...

Added: December 26, 2018

Prokhorov Y., Труды Математического института им. В.А. Стеклова РАН 2019 Т. 307 С. 230-253

Классифицированы некоторые специальные классы трехмерных нерациональных многообразий Фано с терминальными особенностями. В частности, найдены все такие гиперэллиптические и тригональные многообразия. ...

Added: May 10, 2020

Prokhorov Y., Kuznetsov A., Rationality of Mukai varieties over non-closed fields / Cornell University. Series arXiv "math". 2020.

We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus g∈{7,8,9,10} over an arbitrary field 𝗄 of zero characteristic. In the case of dimension n≥4 we prove that these varieties are 𝗄-rational if and only if they have a 𝗄-point except for the case of genus 9, where we assume n≥5. Furthermore, we prove that Mukai varieties of ...

Added: August 19, 2020

Rumynin D., Colloquium Mathematicum 2021 Vol. 164 P. 123-131

We investigate geometry of D-affine varieties. Our main result is that a D-affine rational projective surface over an algebraically closed field is a generalised flag variety of a reductive group. ...

Added: September 7, 2021

Prokhorov Y., Kuznetsov A., Rationality over non-closed fields of Fano threefolds with higher geometric Picard rank / Cornell University. Series arXiv "math". 2021.

We prove rationality criteria over algebraically non-closed fields of characteristic 0 for five out of six types of geometrically rational Fano threefolds of Picard number 1 and geometric Picard number bigger than 1. For the last type of such threefolds we provide a unirationality criterion and prove stable non-rationality under additional assumptions. ...

Added: November 23, 2021

Przyjalkowski V., Shramov K., Труды Математического института им. В.А. Стеклова РАН 2016 Т. 294 С. 167-190

We classify three-dimensional nodal Fano varieties that are double covers of smooth quadrics branched over intersections with quartics acted on by finite simple non-abelian groups, and study their rationality. ...

Added: October 13, 2016

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