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.

# Your search: "author:"Morrison, David R""

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In this work we investigate the global symmetries of six-dimensional superconformal field theories (6D SCFTs) via their description in F-theory. We provide computer algebra system routines determining global symmetry maxima for all known 6D SCFTs while tracking the singularity types of the associated elliptic fibrations. We tabulate these bounds for many CFTs including every 0-link based theory. The approach we take provides explicit tracking of geometric information which has remained implicit in the classifications of 6D SCFTs to date. We derive a variety of new geometric restrictions on collections of singularity collisions in elliptically fibered Calabi-Yau varieties and collect data from local model analyses of these collisions. The resulting restrictions are sufficient to match the known gauge enhancement structure constraints for all 6D SCFTs without appeal to anomaly cancellation and enable our global symmetry computations for F-theory SCFT models to proceed similarly.

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