Deformations and Simultaneous Resolution of Determinantal Surfaces

TL;DR

This paper introduces a method using reconstruction algebras and noncommutative relations to achieve simultaneous resolution of determinantal surfaces, extending classical techniques with deformations and GIT variations.

Contribution

It presents a novel approach combining reconstruction algebras, canonical relations, and deformations for resolving determinantal surfaces simultaneously.

Findings

Constructed simultaneous resolutions using deformed relations.

Integrated noncommutative relations of Ringel's canonical algebra.

Demonstrated the effectiveness of GIT variation in the resolution process.

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.