The goal of this dissertation is to develop tools for studying genus one fibered Calabi-Yau 3-folds without section, with an eye towards applications in F-theory.

In particular:

Locally trivial fibrations are well-understood via the work of Mark Gross and others - we know that there are only finitely many families parametrizing all of them, we know they are classified by the Tate-Shafarevich group and we can use Ogg-Shafarevich theory to classify all such fibrations with a common Jacobian.

Since we have an genus one fibration structure, we can always find an equation describing the generic fiber as a genus one curve in some projective vector bundle. Furthermore, we can obtain an equation for the Jacobian from an equation for the torsor.

The work of Mark Gross is especially relevant and helpful, but it has two shortcomings:

Ogg-Shafarevich theory can only be used to classify fibrations whose geometry is reasonably nice - it requires a smooth base, with a simple normal crossings divisor, and only classifies fibrations without multiple fibers. We will see that there are fibrations which simultaneously fail all of these conditions.

It is important to be able to relate arithmetic phenomena to degenerations on elliptic fibrations in every possible codimension. Purity for the Brauer group means we shouldn't expect to gain precise information about degenerations in codimension 2. The Jacobian formulae can be used to obtain information about codimension 2 singularities by brute force. However, the problem there is that the formulae become too messy to actually be used to analyze codimension 2 singularities.

In Part I and Part II, we go over the necessary background. In Part III, we begin our analysis of torsors. We start by reviewing the theory of torsors of index 2 and 3. We will see that the data of the torsor can be recovered from the data a trace zero point on the Jacobian. In the final chapter, we propose using \emph{trace zero varieties} as a tool for studying torsors in general, and explain why this approach is well suited for answering some outstanding questions in F-theory.

Trace zero points are part of the data used to study torsors in either of the other two settings. If we have a class in $\Sh$, it can be represented by a cocycle, and the cocycle is determined by a set of trace zero points.

Trace zero points are easy to parametrize. In the final chapter, we construct a variety that parametrizes pairs consisting of an elliptic curve and a point of trace zero on that elliptic curve.

Trace zero points can be thought of as a generalizing of torsion points. We have a good understanding of moduli space for torsion pairs, and in fact we will use those moduli spaces to bound torsion on elliptically fibered Calabi-Yau 3-folds.

We will also explain how we hope to use these ideas in future work in the last chapter.

Quantum field theory plays a central role in the study of theoretical physics, where a satisfying understanding of general field theories is usually absent. In this thesis, the path integral approach will be revisited, and it will be shown how to organize the computation of path integral into the framework of (fully) extended field theory. Several aspects of quantum field theory will be explored with this new viewpoint, including symmetries and anomalies.

Motivated by treating the 2 → 1 parity-oblivious multiplexing under the influence of experimental noise, with its success probability serving as an indicator of nonclassicality, this thesis establishes connections between various notions of nonclassicality within the context of what is commonly referred to as the simplest nontrivial scenario (a prepare and measure scenario comprised of four preparations and two binary-outcome tomographically complete measurements). Specifically, we relate the established method developed by Pusey in [20] to witness a violation of preparation noncontextuality, which is not suitable in experiments where the operational equivalences to be tested are specified in advance, with a novel approach based on the notion of bounded ontological distinctness for preparations, defined by Chaturvedi and Saha in [9]. In our approach, we test bounded ontological distinctness for two particular preparations that are relevant in certain information processing tasks in that they are associated with the even- and odd-parity of the bits to be communicated. When there exists an ontological model where this distance is preserved we term this parity preservation. Our main result provides a noise threshold under which violating parity preservation (and so bounded ontological distinctness) agrees with the established method for witnessing preparation contextuality in the simplest nontrivial scenario. This is achieved by first relating the violation of parity preservation to the quantification of contextuality in terms of inaccessible information as developed by Marvian in [17], that we also show, given the way we quantify noise, to be more robust in witnessing contextuality than Pusey’s noncontextuality inequality. As an application of our findings, we treat the case of two-bit parity-oblivious multiplexing in the presence of noise. Specifically, leveraging the identified noise threshold for the existence of preparation contextuality, we establish a condition for which preparation contextuality is present in the case where the probability of success exceeds that achieved by any classical strategy. Overall, our results highlight that, below a certain threshold, all the different methods to witness nonclassicality agree. Consequently, an experimenter can choose the most suitable method based on their specific needs.

One of the main research programs in Algebraic Geometry is the classification of varieties. Towards this goal two methodologies arose, the first is classifying varieties up to isomorphism which leads to the study of moduli spaces and the second is classifying varieties up to birational equivalences which leads to the study of birational geometry. Part of the engine of the birational classification is the Minimal Model Program which, given a variety, seeks to find "nice" birational models, which we call minimal models. Towards this direction much progress has been made but there is also much to be done. One aspect of interests is the role of algebraic fiber spaces as the end results of the Minimal Model Program are categorized into Mori fiber spaces, Iitaka fibrations over canonical models and varieties of general type. A natural problem to consider is, starting with an algebraic fiber space, how might it behave with respect to the Minimal Model Program. For case of elliptic threefolds, it was shown by Grassi, that minimal models of elliptic threefolds relate to log minimal models of the base surface. This shows that minimal models, in a sense, have to respect the fiber structure for elliptic threefolds. In this dissertation, I will provide a framework towards a generalization for higher dimensional elliptic fibration and along the way recover the results of Grassi for elliptic fourfolds with section.