TY - THES
AU - Jiang, Yunfeng
PY - 2007
TI - Toric orbifold Chow rings
KW - Thesis/Dissertation
LA - eng
M3 - Text
AB - 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.
N2 - 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.
UR - https://open.library.ubc.ca/collections/831/items/1.0080418
ER - End of Reference