Deformations and Simultaneous Resolution of Determinantal Surfaces
Brian Makonzi
TL;DR
This work develops a noncommutative framework for the simultaneous resolution of determinantal surface singularities by extending reconstruction algebras with Ringel's canonical relation and then deforming the remaining relations alongside a GIT variation. The authors introduce deformed reconstruction algebras $\Lambda_{\boldsymbol{\upgamma}}$ parametrized by $\boldsymbol{\upgamma}$ in a base $\Delta$, and show that the corresponding moduli spaces $\mathcal{M}_{\upvartheta_0}(\Lambda_{\boldsymbol{\upgamma}},\boldsymbol{\updelta})$ are two-dimensional and smooth, enabling a deformation family whose fibers align with the determinantal singularity via a flat map $\uppi$. A separate invariant representation variety $\EuScript{R}^G$ is analyzed and shown to be generated by explicit cycles, allowing the construction of a smooth morphism $\upphi$ from a moduli space to $\Delta$ and leading to a diagram that provides a simultaneous resolution: $\upphi$ is smooth and $\uppi$ is flat, with the origin fiber recovering $\mathrm{Spec}\,R$. The results extend prior cyclic and Gorenstein cases to broader determinantal singularities and furnish a coherent noncommutative-geometry mechanism for simultaneous resolution via deformed reconstruction algebras and their moduli.
Abstract
This paper uses reconstruction algebras to construct simultaneous resolution of determinantal surfaces. The main new difference to the classical case is that, in addition to the quiver of the reconstruction algebra, certain noncommutative relations, namely those of the canonical algebra of Ringel, are required. All the relations of the reconstruction algebra except the canonical relation are then deformed, and these deformed relations together with variation of the GIT quotient achieve the simultaneous resolution.