| In the first part of this thesis, we study derived categories of toric Deligne-Mumford stacks. Our main result is a construction of full strong exceptional collections of line bundles on smooth toric Fano Deligne-Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that our approach will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne-Mumford stacks. In the second part of the thesis, we construct series of smooth Calabi-Yau threefolds with non-abelian fundamental groups by considering free group quotients on complete intersections of quadrics. |
