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 and a finite collection of non-intersecting positive dimensional algebraic sets , we construct a nonnoetherian coordinate ring whose variety coincides with except that each 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.
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Charlie Beil, Nonnoetherian singularities and their noncommutative blowups. J. Noncommut. Geom. 17 (2023), no. 2, pp. 469–498DOI 10.4171/JNCG/495