### ?

## The Fano variety of lines and rationality problem for a cubic hypersurface

arxiv.org.
math.
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., Twisted cubics and quadruples of points on cubic surfaces / 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., On pairs, triples and quadruples of points on a cubic surface / 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

Colliot-Thélène J., Kunyavskiĭ B., Popov V. 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

Trepalin A., 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

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

Prokhorov Y., Documenta Mathematica 2010 Vol. 15 P. 843-872

We study Q-Fano threefolds of large Fano index. In
particular, we prove that the maximum possible Fano index is attained
only by the weighted projective space P(3,4,5,7). ...

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
Cartier and divisible by 2 with the additional assumption that the G-invariant part of the Weil divisor
class group is of rank 1 with respect to an action of some group G. In particular, we find a lot of
examples of Fano 3-folds with “many” symmetries. ...

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 ...

Added: December 17, 2015

Cheltsov I., Shramov K., 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. ...

Added: January 27, 2014

Prokhorov Y., Advances in Geometry 2013 Vol. 13 No. 3 P. 419-434

We classify Fano threefolds with only Gorenstein terminal singularities and Picard
number greater than 1, satisfying the additional assumption that the G-invariant part of the Weil
divisor class group is of rank 1 with respect to an action of some group G. ...

Added: October 7, 2013

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

Added: December 5, 2013

Iliev A., Katzarkov L., Przyjalkowski V., 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

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., Prokhorov Y., 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., Stable birational invariants with Galois descent and differential forms / 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 ...

Added: October 13, 2020

Prokhorov Y., 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., 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

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 ...

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 ...

Added: February 6, 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

Ananʼin S., Verbitsky M., Journal de Mathématiques Pures et Appliquées 2014 Vol. 101 No. 2 P. 188-197

Let M be a compact hyperkähler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in H^2(M) defines a divisor Dv in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W. ...

Added: January 28, 2015