Nonnoetherian singularities and their noncommutative blowups

TL;DR

This paper introduces a new class of nonnoetherian varieties linked to dimer algebras and demonstrates that their noncommutative blowups serve as desingularizations, expanding the understanding of singularities in nonnoetherian algebraic geometry.

Contribution

It constructs nonnoetherian coordinate rings with specific geometric properties and shows their noncommutative blowups act as desingularizations, a novel approach in the field.

Findings

Constructed nonnoetherian coordinate rings with prescribed geometric features.

Proved noncommutative blowups provide desingularizations of certain singularities.

Linked the theory to the central geometry of dimer algebras.

Abstract

We establish a new fundamental class of varieties in nonnoetherian algebraic geometry related to the central geometry of dimer algebras. Specifically, given an affine algebraic variety $X$ and a finite collection of non-intersecting positive dimensional algebraic sets $Y_i \subset X$, we construct a nonnoetherian coordinate ring whose variety coincides with $X$ except that each $Y_i$ is identified as a distinct positive dimensional closed point. We then show that the noncommutative blowup of such a singularity is a noncommutative desingularization, in a suitable geometric sense.