Mutations of splitting maximal modifying modules: The case of reflexive polygons

Abstract

It is known that every three dimensional Gorenstein toric singularity has a crepant resolution. Although it is not unique, all crepant resolutions are connected by repeating the operation "flop". On the other hand, this singularity also has a non-commutative crepant resolution (= NCCR) which is constructed from a consistent dimer model. Such an NCCR is given as the endomorphism ring of a certain module which we call splitting maximal modifying module. In this paper, we show that all splitting maximal modifying modules are connected by repeating the operation "mutation" of splitting maximal modifying modules for the case of toric singularities associated with reflexive polygons.

50 pages, including many figures, v4: corrected the proof of Lemma 3.7, v3: revised several places, especially subsection 5.6, v2: minor changes

References in corpus (5)

• Brane Tilings
• An introduction to the dimer model
• Brane Tilings and Reflexive Polygons
• A Dimer ABC
• Remarks on divisorial ideals arising from dimer models

Cited by in corpus (3)

• Remarks on divisorial ideals arising from dimer models
• Non-commutative crepant resolutions of Hibi rings with small class group
• Semi-steady non-commutative crepant resolutions via regular dimer models