Dual graphs of exceptional divisors

TL;DR

This paper investigates the topological complexity of dual graphs of exceptional divisors in resolutions of singularities, showing that their homotopy types can be arbitrary and characterizing those from rational singularities.

Contribution

It proves that the homotopy type of dual graphs can be arbitrary and characterizes which types arise from rational singularities.

Findings

Homotopy types of dual graphs can be arbitrary.

Homotopy types from rational singularities are characterized.

Abstract

Let p be a singular point of a variety. Consider a resolution where the preimage of p is a simple normal crossing divisor E. The combinatorial structure of E is described by a cell complex D(E), called the dual graph or dual complex of E. It is known that the homotopy type of D(E) depends only on p, not on the resolution chosen. We prove that this homotopy type can be arbitrary. We also describe which homotopy types can be obtained from rational singularities.