On the geometry of C^3/D_27 and del Pezzo surfaces

TL;DR

This paper explores the detailed geometry of a specific orbifold related to string theory, providing explicit desingularizations and linking it to known algebraic surfaces, thereby enhancing understanding of its structure.

Contribution

It explicitly constructs a desingularization of the orbifold C^3/D_27 and relates it to the total space of the canonical bundle over a quasi del Pezzo surface.

Findings

Realized a map between the orbifold and a del Pezzo related space

Established the relationship between the normalizer group of D_27 and the Hessian group

Analyzed the behavior of the Hesse pencil under the quotient

Abstract

We clarify some aspects of the geometry of a resolution of the orbifold X = C3/D_27, 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 D_27 and the Hessian group and on the study of the behaviour of the Hesse pencil of plane cubic curves under the quotient.