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The Fano variety of lines and rationality problem for a cubic hypersurface
Cornell University
,
2014.
No. 1405.5154.
Galkin S., Shinder E.
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 to a Hilbert scheme of two points on a K3 surface; in particular, general cubic fourfold is irrational.
Keywords: схема Гильбертаиррациональностьrationalityрациональность the rationality hypothesisгипотеза рациональностиHilbert schemeBirational geometryбирациональная геометрияHodge structuresструктуры Ходжамногообразие Фаноcubic polynomialFano varietystable birational invariantsстабильные бирациональные инвариантыгиперповерхностиK3 surfacesirrationalityК3 поверхностьcubic hypersurfaceGrothendieck ring of varietiesvariety of linesкубическая гиперповерхностькольцо Гротендика многообразиймногообразие прямых
Popov P., / Cornell University. Series math "arxiv.org". 2018. No. 1810.04563.
We study relations in the Grothendieck ring of varieties which connect the Hilbert scheme of points on a cubic hypersurface Y with a certain moduli space of twisted cubic curves on Y. These relations are generalizations of the "beautiful" Y-F(Y) relation by Galkin and Shinder which connects Y with the Hilbert scheme of two points on Y and the Fano variety F(Y) of lines on Y. We ...
Added: October 23, 2018
Galkin S., Popov P., / Cornell University. Series math "arxiv.org". 2018. No. 1810.07001.
Let X(n) denote n-th symmetric power of a cubic surface X. We show that X(4)×X is stably birational to X(3)×X, despite examples when X(4) is not stably birational to X(3). ...
Added: October 19, 2018
Prokhorov Y., Kuznetsov A., Shramov K., Japanese Journal of Mathematics 2018 Vol. 13 No. 1 P. 109-185
We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds of Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold ...
Added: November 22, 2017
Prokhorov Y., Sbornik Mathematics 2013 Vol. 204 No. 3 P. 347-382
We classify $\mathbb Q$-Fano threefolds of Fano index > 2 and sufficiently big degree. ...
Added: October 7, 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
Latypov I., Социологические исследования 2022 № 5 С. 3-14
Counterfinality is the unintended consequences stemming from uncoordinated actions.
It may be described as a complex theoretical problem, including understanding the motive for action,
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Added: June 9, 2022
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We study Q-Fano threefolds of large Fano index. In
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Added: December 6, 2013
Prokhorov Y., Advances in Geometry 2013 Vol. 13 No. 3 P. 389-418
We classify Fano threefolds with only terminal singularities whose canonical class is
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Added: October 7, 2013
Vologodsky V., Finkelberg M. V., Bezrukavnikov R., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163-190
Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...
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Cheltsov Ivan, Shramov Constantin, Experimental Mathematics 2013 Vol. 22 No. 3 P. 313-326
We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose α-invariant of Tian is greater than 2/3. ...
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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
Boston : Birkhäuser, 2013
This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the ...
Added: February 14, 2013
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Prokhorov Y., Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 215-229
We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds. ...
Added: January 24, 2014
Kishimoto T., Yuri Prokhorov, Zaidenberg M., Osaka Journal of Mathematics 2014 Vol. 51 No. 4 P. 1093-1113
We address the following question: When an affine cone over a smooth Fano threefold admits an effective action of the additive group? In this paper we deal with Fano threefolds of index 1 and Picard number 1. Our approach is based on a geometric criterion from our previous paper, which relates the existence of an ...
Added: October 10, 2013
Rovinsky M., / Cornell University. Series math "arxiv.org". 2012.
I show that the cohomology of the generic points of algebraic complex varieties becomes stable birational invariant, when considered `modulo the cohomology of the generic points of the affine spaces'. ...
Added: October 31, 2013
Przyjalkowski V., Shramov K., Communications in Number Theory and Physics 2020 Vol. 14 No. 3 P. 511-553
We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension n as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed n+1. Based on this bound ...
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Ю. Г. Прохоров, Известия РАН. Серия математическая 2013 Т. 77 № 3 С. 199-222
We study elements $\tau$ of order two in the birational automorphism groups of rationally connected three-dimensional algebraic varieties such that there exists a non-uniruled divisorial component of the $\tau$-fixed point locus. Using the equivariant minimal model program, we give a rough classification of such elements. ...
Added: July 1, 2013
Gusein-Zade S., Proceedings of the Edinburgh Mathematical Society 2019 Vol. 62 No. 4 P. 925-948
We define a Grothendieck ring of varieties with actions of finite groups and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural λ-structures on the ring and the corresponding power structures over it and ...
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Gusein-Zade S., Revista Matemática Complutense 2018 Vol. 31 No. 3 P. 595-609
A power structure over a ring is a method to give sense to expressions of the form $(1+a_1t+a_2t^2...)^m$, where $a_i$, $i=1,2, ...$, and $m$ are elements of the ring. The (natural) power structure over the Grothendieck ring of complex quasi-projective varieties appeared to be useful for a number of applications. We discuss new examples of ...
Added: October 27, 2020
Verbitsky M., Geometry and Topology 2014 Vol. 18 No. 2 P. 897-909
A Hermitian metric ω on a complex manifold is called SKT or pluriclosed if ddcω=0. Let M be a twistor space of a compact, anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is Kähler, hence isomorphic to CP3 or a flag space. This result is obtained from rational ...
Added: April 29, 2014
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 ...
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Prokhorov Y., Advances in Geometry 2013 Vol. 13 No. 3 P. 419-434
We classify Fano threefolds with only Gorenstein terminal singularities and Picard
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Added: October 7, 2013
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
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Added: March 17, 2013