Lines on Cubic Threefolds and Fourfolds Containing a Plane
| dc.contributor.advisor | Addington, Nicolas | |
| dc.contributor.author | Brooke, Corey | |
| dc.date.accessioned | 2024-01-09T22:49:04Z | |
| dc.date.available | 2024-01-09T22:49:04Z | |
| dc.date.issued | 2024-01-09 | |
| dc.description.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 $k=\bC$. | en_US |
| dc.identifier.uri | https://hdl.handle.net/1794/29176 | |
| dc.language.iso | en_US | |
| dc.publisher | University of Oregon | |
| dc.rights | All Rights Reserved. | |
| dc.title | Lines on Cubic Threefolds and Fourfolds Containing a Plane | |
| dc.type | Electronic Thesis or Dissertation | |
| thesis.degree.discipline | Department of Mathematics | |
| thesis.degree.grantor | University of Oregon | |
| thesis.degree.level | doctoral | |
| thesis.degree.name | Ph.D. |
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