Center for Theoretical Physics of the Universe (CTPU-PTC)

Birational Models of Elliptic Fibrations

by David Wen (NTU)

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Elliptic fibrations are varieties which admit a fiber structure whose general fibers are elliptic curves. They are frequently considered test cases in algebraic geometry as a special case of fibrations of varieties with trivial canonical divisors (e.g. fibration of Calabi-Yau varieties). This is of interest in the study of Calabi-Yau varieties since if a Calabi-Yau variety admits a non-trivial fibration structure, then it necessarily has to be a fibration of lower dimensional varieties with trivial canonical divisors. Thus, these fibrations are geometric structures that allow for study of the variety itself through studying the bases and the fibers of the fibrations. This idea is of particular importance to F-theory, which uses elliptically fibered Calabi-Yau varieties as mathematical models to study aspects of String Theory. We have that case of 6D F-theory is well studied due in part to the study of elliptic Calabi-Yau threefolds, yet moving on to the case of 4D F-theory is more difficult since this requires a mathematical understanding of elliptic Calabi-Yau fourfolds, much of which is not as well developed or as well behaved as in the threefold case. In this talk, I will discuss the problems of generalizing the birational aspects of elliptic Calabi-Yau threefold cases to the situation of elliptic Calabi-Yau fourfolds.