Branes at ℂ4/F singularity from toric geometry

We study toric singularities of the form of ℂ4/T for finite abelian groups Γ ⊂ SU(4). In particular, we consider the simplest case Γ = ℤ₂ × ℤ₂ × ℤ₂ and find explicitly charge matrices for partial resolutions of this orbifold by extending the method by Morrison and Plesser. We obtain three kinds of algebraic equations, z₁z₂z₃z₄ = z₅², z₁z₂z₃ = z₄²z₅ and z₁z₂z₅ = z₃z₄ where z_i's parametrize ℂ⁵. When we put N D1 branes at this singularity, it is known that the field theory on the worldvolume of N D1 branes is T-dual to 2 × 2 × 2 brane cub model. We analyze geometric interpretation for field theory parameters and moduli space.