The blow-up of a singularity at the module of derivations
Paul Barajas, Enrique Chávez-Martínez, Agustín Romano-Velázquez
TL;DR
The paper investigates resolving singularities by iteratively blowing up the module of derivations $\mathrm{Der}_{\mathbb C}(\mathcal O_X)$. It proves a Nobile-type criterion: if the blow-up $\mathrm{Bl}_{\mathrm{Der}}(X)\to X$ is an isomorphism under the Zariski–Lipman conjecture, then $X$ is smooth, and it establishes a positive resolution result for complex analytic log-canonical surface singularities by finite sequences of such blow-ups. The authors carry out explicit analysis for rational and $A_n$-type surface singularities, and treat generalized simple elliptic and cusp cases via the structure of reflexive modules and the minimal adapted resolution, leveraging McKay-type perspectives. The work highlights a canonical alternative to Nash blow-ups for resolution and demonstrates termination through contraction of divisors and the geometry of full/sheaf-theoretic data. The results collectively show that, in key classes (curves, rational and log-canonical surfaces), blow-ups along derivations canonically resolve singularities, delineating both method and limitations relative to higher dimensions and the Zariski–Lipman framework.
Abstract
We study the problem of resolving singularities via the blow-up of the module of derivations. Our main results are a positive answer for the case of curves and log-canonical surface singularities, i.e., a finite sequence of blow-ups along the module of derivations resolves the singularities of such varieties.