On Tachyons in Generic Orbifolds of BC^r and Gauged Linear Sigma Models

We study some aspects of Gauged Linear Sigma Models corresponding to orbifold singularities of the form $\BC^r/\Gamma$, for $r=2,3$ and $\Gamma = \BZ_n$ and $\BZ_n\times \BZ_m$. These orbifolds might be tachyonic in general. We compute expressions for the multi parameter sigma model Lagrangians for these orbifolds, in terms of their toric geometry data. Using this, we analyze some aspects of the phases of generic orbifolds of $\BC^r$.