Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Mathematics

Supervisor

Dhillon, Ajneet

Abstract

This thesis explores two problems in algebraic geometry concerning derived categories and essential dimension. In the first part, we use conservative descent to study semi-orthogonal decompositions for some homogeneous varieties over general bases. We produce a semi-orthogonal decomposition for the derived category of coherent sheaves on a generalized Severi-Brauer scheme over an algebraic stack. This extends known results for Severi-Brauer varieties and Grassmanianns. We use our results to construct semi-orthogonal decompositions for flag varieties over arbitrary bases. \\ In the second part, we compute the essential dimension of the moduli stack $\Bun_{\Sp_n}$ of symplectic bundles over a smooth projective curve $X$ of genus $g \ge 2$. The notion of essential dimension of an algebraic object introduced by J.Buhler and Z.Reichstein is the minimal number of algebraically independent objects needed to parameterize the object. For the moduli stack of symplectic vector bundles, the essential dimension naturally splits into two components. The first component is the essential dimension of the residual gerbe, which we compute by relating it to the essential dimension of hermitian modules. The second component involves determining the transcendence degree of the field of moduli. We address this by estimating the dimension of the moduli stack of nilpotent skew-symmetric endomorphisms, using deformation theory techniques.

Summary for Lay Audience

The thing that has most fascinated me about mathematics are the deep connections that exists between seemingly very unrelated fields, fields that were developed to address very different problems. This thesis deals with two such fields, which have developed intricate links over time. On one side is \emph{algebraic geometry}, the study of geometric spaces through the algebra of functions defined on them. On the other side is \emph{group theory}, which explores symmetry and how it governs the properties of mathematical objects. The properties of a geometric space with symmetry given by a specific group $G$, captures the complexity of the group. The objective of this thesis is to understand the complexity of this group $G$ by applying algebro-geometric techniques to such geometric spaces. The first part of my thesis focuses on \emph{derived category of sheaves} on a space. Sheaves on a space $X$ can be thought of as linear spaces parametrized by the base space $X$. Derived categories of sheaves on the other hand is a systematic method to keep track of the complex structures that can arise out of sheaves on $X$. Surprisingly, many properties of $X$ can be recovered by studying the derived category on $X$. My main contribution was constructing special building blocks for the derived categories of certain geometric spaces equipped with specific symmetries given by a group $G$. These building blocks are unsurprisingly constructed by understanding the representations of the group $G$. The second part of my research focuses on \emph{essential dimension}, an invariant that studies the complexity of mathematical objects. This invariant is particularly interesting in the context of a moduli space, as it measures the obstruction to the moduli space being described by a simpler geometric object. The moduli space considered is the moduli space of \emph{symplectic vector bundles} over a curve $X$. These are linear spaces parameterized by the curve $X$ that has a special kind of symmetry. We prove that the difference between the essential dimension of the moduli space and that of the candidate simpler geometric space is accounted for by the essential dimension of the symmetries at each point.

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