A note on cubic fourfolds containing several planes

We study the geometry, Hodge theory and derived category of cubic fourfolds containing several planes and their associated twisted K3 surfaces. We focus on the case of two planes intersecting along a line.

💡 Research Summary

This paper presents a detailed study of the geometry, Hodge theory, and derived categories of smooth cubic fourfolds that contain several planes, with a particular focus on the case where two distinct planes intersect along a line.

The starting point is the well-established association, due to Kuznetsov and refined by Moschetti, between a cubic fourfold X containing a plane P and a twisted K3 surface (S_P, α_P). There exists a Fourier-Mukai equivalence between the Kuznetsov component A_X of D^b(X) and the derived category of this twisted K3 surface, D^b(S_P, α_P). Consequently, if X contains two planes P1 and P2, one obtains two such twisted K3 surfaces, and the paper investigates their interrelations.

The central geometric scenario is when P1 and P2 intersect along a line L. For a very general such cubic fourfold, the paper establishes several key results:

1. Non-isomorphism of K3 surfaces: The associated (untwisted) K3 surfaces S_{P1} and S_{P2} are not isomorphic. This is proven by degenerating to the special case of Eckardt cubic fourfolds, where the associated plane sextic curves can be shown to be non-isomorphic. Consequently, S_{P1} and S_{P2} are not even derived equivalent.
2. Geometric Correspondence via F_L: The core construction is a smooth surface F_L, which serves as a correspondence between S_{P1} and S_{P2}. This surface is obtained from the locus of lines in X that intersect L. It admits two distinct double covers f_i: F_L → S_{P_i} and an elliptic fibration F_L → L. These maps fit into a commutative diagram involving the Gauss map of X restricted to L, providing a concrete geometric bridge.
3. Hodge-theoretic Interpretation: The Fano correspondence induces a Hodge isometric embedding of the transcendental lattice T(X) of the cubic fourfold into the transcendental lattice of each S_{P_i}. This embedding is reinterpreted via the surface F_L: the lattice T(X) is shown to be Hodge isometric to the intersection of the pullbacks f1T(S_{P1}) and f2T(S_{P2}) inside H^2(F_L, Z), up to a scaling factor.
4. Elliptic Fibration Structure: The K3 surfaces S_{P_i} are shown to admit elliptic fibrations whose Jacobians are isomorphic to a common elliptic K3 surface S. Furthermore, S_{P1} and S_{P2} are realized as Tate–Šafarevič twists of this common S, with the Brauer classes being exchanged under the correspondence.
5. Derived Categorical Realization: The geometric correspondence is lifted to the level of derived categories. The surface F_L comes with two involutions ι_i whose quotient is S_{P_i}. The derived category D^b(F_L) admits semiorthogonal decompositions related to its equivariant categories with respect to these involutions. Specifically, the equivariant category D^b(F_L)^{Φ_i} (for an autoequivalence Φ_i of order two) decomposes as ⟨ D^b(S_{P_i}, α_{P_i}), D^b(E) ⟩, where E is the branch locus. Moreover, there exists an equivalence between these two equivariant categories that respects these decompositions, offering a categorical explanation for the twisted derived equivalence between D^b(S_{P1}, α_{P1}) and D^b(S_{P2}, α_{P2}).

The paper also briefly discusses the cases where the two planes are disjoint or intersect in a point. The analysis combines classical algebraic geometry, the theory of periods and Hodge structures for K3 surfaces and cubic fourfolds, and modern techniques from derived categories, providing a comprehensive picture of the rich structures arising from this specific geometric configuration.