Brian Makonzi | Searcharxiv

arXiv2023

Deformations and Simultaneous Resolution of Determinantal Surfaces

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.

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arXiv2022

The Artin Component and Simultaneous Resolution via Reconstruction Algebras of Type $A$

This paper uses noncommutative resolutions of non-Gorenstein singularities to construct classical deformation spaces, by recovering the Artin component of the deformation space of a cyclic surface singularity using only the quiver of the corresponding reconstruction algebra. The relations of the reconstruction algebra are then deformed, and the deformed relations together with variation of the GIT quotient achieve the simultaneous resolution. This extends work of Brieskorn, Kronheimer, Grothendieck, Cassens-Slodowy and Crawley-Boevey-Holland into the setting of singularities $\mathbb{C}^2/H$ with $H\leq\mathrm{GL}(2,\mathbb{C})$, and furthermore gives a prediction for what is true more generally.

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