On the exceptional set of crepant resolutions of abelian singularities
Ugo Bruzzo, Fábio Arceu Ferreira
TL;DR
The paper investigates abelian quotient singularities $\mathbb{C}^{n}/G$ with $G\subset SL(n,\mathbb{C})$, focusing on toric realizations and crepant resolutions via Hilbert basis methods. It establishes that for each junior class $g$ of $G$, the associated exceptional divisor $E_g$ admits an open toric neighborhood isomorphic to the total space of a line bundle on $E_g$, and, in the crepant case, $E_g$ is normally embedded with the neighborhood modeled by the total space of $\omega_{E_g}$. In dimension $3$, the crepant toric resolution can be glued from the total spaces $\operatorname{tot}(\omega_{E_g})$ corresponding to compact junior elements, providing explicit local-to-global descriptions of the resolution. A key theme is the interplay between combinatorial data (Hilbert basis, junior simplex) and geometric structures (tubular neighborhoods, normal bundles), yielding a concrete partial solution to the classical neighborhood-of-zero-section problem in algebraic geometry. The results connect the McKay correspondence, toric minimal models, and deformation theory to give a cohesive toric framework for understanding local neighborhoods around exceptional divisors.
Abstract
Let G be a finite abelian subgroup of SL(n,C), and suppose there exists a toric crepant resolution phi: X -- > C^n/G. We prove that for each component E of the exceptional set of phi there exists an open subset U of X that contains E and is isomorphic to the total space of the canonical bundle of E. This contributes to the collection of results aimed at solving a classical problem, i.e., to determine which submanifolds of a complex manifold have a neighborhood isomorphic to a neighborhood of the zero section of their normal bundle.