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Twisted forms of toric varieties, their derived categories, and their rationality McFaddin, Patrick


Toric varieties defined over the complex numbers provide an important testing ground for computing various algebro-geometric invariants (e.g., the coherent derived category associated to a variety), as many computations of interest may be phrased entirely in terms of combinatorial data such as fans, cones, polytopes. Over general fields, we consider twisted forms of such objects called "arithmetic toric varieties", whose analysis is naturally Galois-theoretic. In this talk, we will present results on the structure of derived categories of arithmetic toric varieties via exceptional collections. In particular, we will focus on some highly symmetric classes of such objects, including centrally symmetric toric Fano varieties and toric varieties associated to root systems of type A. The latter class yields examples of varieties which are arithmetically interesting but whose derived categories are well understood. A conjecture of Orlov posits that the structure of the derived category influences the rationality type of a variety. We will discuss how this plays out in the setting of toric varieties. This is joint work with Matthew Ballard, Alexander Duncan, and Alicia Lamarche.

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