Mathematics Theses and Dissertations

Permanent URI for this collection

https://hdl.handle.net/1794/2044

This collection contains some of the theses and dissertations produced by students in the University of Oregon Mathematics Graduate Program. Paper copies of these and other dissertations and theses are available through the UO Libraries.

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Browsing Mathematics Theses and Dissertations by Author "Addington, Nicolas"

Now showing 1 - 4 of 4
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    A Categorical sl_2 Action on Some Moduli Spaces of Sheaves
    (University of Oregon, 2020-09-24) Takahashi, Ryan; Addington, Nicolas
    We study a certain sequence of moduli spaces of stable sheaves on a K3 surface of Picard rank 1 over $\mathbb{C}$. We prove that this sequence can be given the structure of a geometric categorical $\mathfrak{sl}_2$ action, a global version of an action studied by Cautis, Kamnitzer, and Licata. As a corollary, we find that the moduli spaces in this sequence which are birational are also derived equivalent.
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    Lines on Cubic Threefolds and Fourfolds Containing a Plane
    (University of Oregon, 2024-01-09) Brooke, Corey; Addington, Nicolas
    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 $k=\bC$.
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    Moduli Spaces of Hermite-Einstein Connections over K3 Surfaces
    (University of Oregon, 2020-09-24) Wray, Andrew; Addington, Nicolas
    We study the moduli space M of twisted Hermite-Einstein connections on a vector bundle over a K3 surface X. We show that the universal bundle can be viewed as a family of stable vector bundles over M parameterized by X, therefore identifying X with a component of a moduli space of sheaves over M. The proof hinges on a new realization of twisted differential geometry that puts untwisted and twisted bundles on equal footing. Moreover, we use this technique to give a new and streamlined proof that M is nonempty, compact, and deformation-equivalent to a Hilbert scheme of points on a K3 surface, and that the Mukai map is a Hodge isometry.
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    Moduli Spaces of Sheaves on K3 Surfaces and Galois Representations
    (University of Oregon, 2019-09-18) Frei, Sarah; Addington, Nicolas
    We consider two K3 surfaces defined over an arbitrary field, together with a smooth proper moduli space of stable sheaves on each. When the moduli spaces have the same dimension, we prove that if the etale cohomology groups with $\Q_\ell$ coefficients of the two surfaces are isomorphic as Galois representations, then the same is true of the two moduli spaces. In particular, if the field of definition is finite and the K3 surfaces have equal zeta functions, then so do the moduli spaces, even when the moduli spaces are not birational. This generalizes works of Mukai, O'Grady, and Markman, who have studied these moduli spaces of sheaves defined over the complex numbers.