The Geometry of the Quotient Stack Arising from a Stacky Fan

A quotient stack, [Z/G], is a geometric object that models the quotient of a space, Z, by the action of a Lie group, G, while carrying additional structure at the singularities. Quotient stacks generalize toric varieties, and thus constitute a broad and important class of geometric spaces. In this dissertation, we will exploit a construction given by Borisov, Chen, and Smith that allows one to construct a quotient stack from a particular combinatorial object, called a stacky fan. Our program is to deduce geometric features of the quotient stack from the stacky fan.;Our main results are to determine the component group of the Lie group G from the combinatorics of the stacky fan. In particular, we will give a necessary and sucient condition on the stacky fan for the corresponding group G to be connected. We will also give a characterization of all the inertia groups of the quotient stack, in terms of the combinatorics of the stacky fan.;Finally, we will turn our attention to the stacky fans that give rise to weighted projective spaces (and fake weighted projective spaces), a very important class of toric varieties. In particular, we will give a characterization of stacky fans that correspond to weighted projective spaces. In the case of 2-dimensional sheared simplices (a special case of the labeled polytopes of Lerman-Tolman), we give an explicit and complete description of the resulting quotient stack, in terms of the greatest common divisor of positive integers associated to the polytope.