Fano varieties of cubic fourfolds containing a plane
type: Item
instance of: scholarly article
title: Fano varieties of cubic fourfolds containing a plane
zbMATH Open document ID: 1266.18016
author: Emanuele Macrì
author: Paolo Stellari
published in: Mathematische Annalen
publication date: 2013-02-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/0909.2725
review text: Since \textit{A. Kuznetsov} conjectured [Prog. Math. 282, 219--243 (2010; Zbl 1202.14012)] that a cubic fourfold \(Y\) is rational if and only if a certain associated subcategory \(\mathbf{T}_Y \subset {D^b}(Y)\) is equivalent to \({D^b}(S)\) for some \(K3\) surface \(S\), much attention has been focussed on understanding such subcategories, and their geometric significance. This paper is concerned with cubic fourfolds \(Y\) containing a plane: in this case, there is a known geometric construction associating a particular \(K3\) surface \(S_Y\) to \(Y\), and Kuznetsov shows (ibid., Theorem 4.3) that \(\mathbf{T}_Y \cong {D^b}(S_Y,\beta)\), the derived category of \(\beta\)-twisted sheaves on \(S_Y\). \textit{A. Kuznetsov} and \textit{D. Markushevich} [J. Geom. Phys. 59, No. 7, 843--860 (2009; Zbl 1181.14049), Section 5] show that the Fano variety \(F(Y)\) of lines on \(Y\) occurs as a moduli space of stable sheaves in \(\mathbf{T}_Y\). Macri and Stellari's first main result (Theorem 1.3) shows that, for generic \(Y\), the Fano variety \(F(Y)\) may also be realized as a moduli space of stable objects for certain explicit Bridgeland stability conditions on \({D^b}(S_Y,\beta)\). Given this background, it is natural to ask whether the rationality of \(Y\) might be addressed via the geometry of the associated Fano variety \(F(Y)\). In this direction, the authors show (Theorem 4.2) that, for cubic fourfolds \(Y\) containing a plane, Kuznetsov's condition holds precisely when \(F(Y)\) is birational to a moduli space of (untwisted) stable sheaves on \(S\). Furthermore, they find (Theorem 1.4) that, for all cubic fourfolds \(Y\) containing a plane, \(F(Y)\) is birational to a smooth projective moduli space of twisted sheaves on a \(K3\) surface obtained as a deformation of \(S_Y\). The twisting here is non-trivial for generic \(Y\): this is in accordance with the expectation that such \(Y\) are non-rational. The moduli spaces appearing in Theorems 1.3 and 1.4 are related by …
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14E08
zbMATH DE Number: 6134306
zbMATH Keywords: derived categories
zbMATH Keywords: cubic fourfolds
zbMATH Keywords: Bridgeland stability
MaRDI profile type: Publication
OpenAlex ID: W2088567882
arXiv ID: 0909.2725
cites work: Bridgeland-stable moduli spaces for \(K\)-trivial surfaces
cites work: Variétés de Prym et jacobiennes intermédiaires
cites work: Variétés kähleriennes dont la première classe de Chern est nulle
cites work: Q3737565
cites work: A categorical invariant for cubic threefolds
cites work: Stability conditions on \(K3\) surfaces
cites work: Stability conditions on triangulated categories
cites work: Twisted Fourier-Mukai functors
cites work: Some remarks on Brauer groups of \(K3\) surfaces
cites work: Tilting in abelian categories and quasitilted algebras
cites work: Q4699490
cites work: Q4946819
cites work: Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces
cites work: Transcendental obstructions to weak approximation on general \(K3\) surfaces
cites work: Q5470192
cites work: The Kähler cone of a compact hyperkähler manifold
cites work: Stability conditions for generic K3 categories
cites work: Equivalences of twisted K3 surfaces
cites work: Q3438052
cites work: Moduli spaces of irreducible symplectic manifolds
cites work: Smoothness of the moduli space of complexes of coherent sheaves on an Abelian or a projective K3 surface
cites work: Derived Categories of Cubic Fourfolds
cites work: Derived categories of quadric fibrations and intersections of quadrics
cites work: ABEL–JACOBI MAPS FOR HYPERSURFACES AND NONCOMMUTATIVE CALABI–YAU'S
cites work: Symplectic structures on moduli spaces of sheaves via the Atiyah class
cites work: On the monodromy of moduli spaces of sheaves on K3 surfaces
cites work: INTEGRAL CONSTRAINTS ON THE MONODROMY GROUP OF THE HYPERKÄHLER RESOLUTION OF A SYMMETRIC PRODUCT OF A K3 SURFACE
cites work: Q3871006
cites work: INTEGRAL SYMMETRIC BILINEAR FORMS AND SOME OF THEIR APPLICATIONS
cites work: Moduli stacks and invariants of semistable objects on \(K3\) surfaces
cites work: Mapping class group and a global Torelli theorem for hyperkähler manifolds
cites work: Théorème de Torelli pour les cubiques de \({\mathbb{P}}^ 5\). (Torelli theorem for the cubics of \({\mathbb{P}}^ 5)\)
cites work: Q3438051
DOI: 10.1007/S00208-011-0776-7
description: scientific article