(0,2)-deformations and the G-Hilbert scheme

We study first order deformations of the tangent sheaf of resolutions of Calabi-Yau threefolds that are of the form C-3/Z(r), focusing on the cases where the orbifold has an isolated singularity. We prove a lower bound on the number of deformations for any crepant resolution of this orbifold. We show that this lower bound is achieved when the resolution used is the G-Hilbert scheme, and note that this lower bound can be found using methods from string theory. These methods lead us to a new way to construct the G-Hilbert scheme using the singlet count.