Lines on Cubic Threefolds and Fourfolds Containing a Plane
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Date
2024-01-09
Authors
Brooke, Corey
Journal Title
Journal ISSN
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Publisher
University of Oregon
Abstract
This thesis describes the Fano scheme $F(Y)$ of lines on a general cubic threefold $Y$ containing a plane over a field $k$ of characteristic different from $2$. One irreducible component of $F(Y)$ is birational (over $k$) to a torsor $T$ of an abelian surface, and we apply the geometry and arithmetic of this torsor to answer two questions. First, when is a cubic threefold containing a plane rational over $k$, and second, how can one describe the rational Lagrangian fibration from the Fano variety of lines on a cubic fourfold containing a plane? To answer the first question, we apply recently developed intermediate Jacobian torsor obstructions and show that the existence over $k$ of certain classical rationality constructions completely determines whether the threefold is rational over $k$. The second question, motivated by hyperkähler geometry, we answer by giving an elementary construction that works over a broad class of base fields where hyperkähler tools are not available; moreover, we relate our construction to other descriptions of the rational Lagrangian fibration in the case $k=\bC$.