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## Rationality over non-closed fields of Fano threefolds with higher geometric Picard rank

Cornell University
,
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.

Publication based on the results of:

Prokhorov Y., / 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., / 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

Popov V., / 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

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

Przyjalkowski V., Cheltsov I., Shramov K., Izvestiya. Mathematics 2019 Vol. 83 No. 4 P. 860-907

We classify smooth Fano threefolds with infinite automorphism groups. ...

Added: August 12, 2020

Vladimir L. Popov, / 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

Prokhorov Y., / Cornell University. Series arXiv "math". 2021.

We obtain a sufficient condition for a Fano threefold with terminal singularities to have a conic bundle structure. ...

Added: November 27, 2021

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

Loginov K., Avilov A., Przyjalkowski V., / Cornell University. Series arXiv "math". 2023.

We study the coregularity of smooth Fano threefolds. We prove that for 100 out of 105 families of smooth Fano threefolds, a general member in the family has coregularity 0; moreover, for 92 families out of these 100, any member in the family has coregularity 0; for the remaining 5 families, we obtain some partial ...

Added: November 7, 2023

Cheltsov I., Kishimoto T., Dubouloz A., / Cornell University. Series arXiv "math". 2020.

We study toric G-solid Fano threefolds that have at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups. ...

Added: August 19, 2020

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

Poddiakov A., / Social Science Research Network. Series SSRN Working Paper Series "SSRN Working Paper Series". 2023.

An important and interesting phenomenon of the last few decades is the increasing number of mathematical studies of so-called intransitive dice with non-standard numbers on their faces and the popularization of them. The dice beat one another like in the rock-paper-scissors game. They violate the transitivity law (or axiom): “if it were true that whenever ...

Added: February 23, 2023

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

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

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

Kuznetsov A., Prokhorov Y., Épijournal de Géométrie Algébrique 2018 Vol. 2 No. 3 P. 1-14

We give an explicit construction of prime Fano threefolds of genus 12 with a Gm-action, describe their isomorphism classes and automorphism groups. ...

Added: September 6, 2018

Kang M., Yuri Prokhorov, Journal of Algebra 2010 Vol. 324 No. 9 P. 2166-2197

Added: December 5, 2013

Prokhorov Y., Kuznetsov A., / 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

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

We prove that Q-Fano threefolds of Fano index ≥8 are rational. ...

Added: June 8, 2019

Galkin S., Shinder E., / 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

Cheltsov I., Shramov K., Przyjalkowski V., / Cornell University. Series arXiv "math". 2019.

We classify smooth Fano threefolds with infinite automorphism groups. ...

Added: November 19, 2019

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

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

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020