Special AAG Seminar: Indecomposable cycles on Cubic fourfolds containing a plane

Abstract: In contrast to the well-known result that a general smooth cubic threefold in $\mathbb{P}^4$ is irrational, the irrationality of cubic fourfolds in $\mathbb{P}^5$ is still open. When the given cubic fourfold is special, one can consider its associated (noncommutative) $K3$ surface in terms of the derived category, and in some cases, there is a specific construction of a corresponding geometric $K3$ surface with a twist. Via the homological projective duality, hence the mirror symmetry can appear here.

In this talk, we focus on the case that the cubic fourfold contains exactly one plane. We introduce a construction of a family ${X_t}$ of cubic fourfolds containing the same plane and exhibit that its base locus should be an important subject from both the rationality problem and the mirror symmetry perspectives. More specifically, we construct a family of indecomposable cycles in $CH^3_{\rm ind}(X_t, 1)$, which are supported on the base locus, and show that it is in the 2-torsion part. This can be considered as a family of 2-torsion Brauer class of $X_t$ by a result of Bruno Kahn, and we expect that it represents the irrationality of the general $X_t$ geometrically.

Host: Matt Kerr