On the geometry of C^3/\Delta_{27} and del Pezzo surfaces

We clarify some aspects of the geometry of a resolution of the orbifold X = ℂ3/Δ27, the noncompact complex manifold underlying the brane quiver standard model recently proposed by Verlinde and Wijnholt. We explicitly realize a map between X and the total space of the canonical bundle over a degree 1 quasi del Pezzo surface, thus defining a desingularization of X. Our analysis relys essentially on the relationship existing between the normalizer group of Δ27 and the Hessian group and on the study of the behaviour of the Hesse pencil of plane cubic curves under the quotient.