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Toric orbifold Chow rings Jiang, Yunfeng

Abstract

In this thesis we study the orbifold Chow ring of smooth Deligne-Mumford stacks which are related to toric models. We give a new quotient construction of toric Deligne-Mumford stacks defined by Borisov-Chen-Smith such that toric Deligne-Mumford stacks have more representations as quotient stacks. We define toric stack bundles using this new construction and compute their orbifold Chow rings. As an interesting application, we compute the orbifold Chow ring of finite abelian gerbes over smooth schemes. The extended stacky fans we introduced are used to give a new quotient construction of toric Deligne-Mumford stacks. These new combinatorial data have relations to stacky hyperplane arrangements, i.e. every stacky hyperplane arrangement determines an extended stacky fan. The hyperplane arrangement determines the topology of the associated hypertoric varieties. We define hypertoric Deligne- Mumford stacks using stacky hyperplane arrangements, generalizing the construction of Hausel and Sturmfels. Their orbifold Chow rings are computed as well. Borisov, Chen and Smith computed the orbifold Chow ring of projective toric Deligne-Mumford stacks. We generalize their formula to semi-projective toric Deligne-Mumford stacks. The hypertoric Deligne-Mumford stack is a closed substack of the Lawrence toric Deligne-Mumford stack associated to the stacky hyperplane arrangement which is semi-projective, but not projective. We prove that the orbifold Chow ring of a Lawrence toric Deligne-Mumford stack is isomorphic to the orbifold Chow ring of its associated hypertoric Deligne-Mumford stack. This is the orbifold Chow ring analogue of a result of Hausel and Sturmfels.

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