Fano varieties of cubic fourfolds containing a plane
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 a particular wall-crossing in the space of Bridgeland stability conditions.
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