Flops and mutations for crepant resolutions of polyhedral singularities

Abstract

Abstract. Let G be a polyhedral group G ⊂ SO(3) of types Z/nZ, D2n and T. We prove that there exists a one-to-one correspondence between flops of G-Hilb(C 3 ) and mutations of the McKay quiver with potential which do not mutate the trivial vertex. This correspondence provides two equivalent methods to construct every projective crepant resolution for the singularities C 3 /G, which are constructed as moduli spaces MC of quivers with potential for some chamber C in the space Θ of stability conditions. In addition, we study the relation between the exceptional locus in MC with the corresponding quiver QC, and we describe explicitly the part of the chamber structure in Θ where every such resolution can be found.

Contractions and deformations

“…We have already observed that this is infinite dimensional, so g contracts a divisor by 4.8. Alternatively, we can see that g contracts a divisor by using the explicit open cover as in [NS,p40]. On the other hand, by general theory (see e.g.…”

Crepant Resolutions of 3-Dimensional Quotient Singularities via Cox Rings

“…Another successful approach to investigating such resolutions (which can be considered an extension of G-Hilbert schemes) goes via quiver representation theory, see e.g. [7,32,30]. However, up to now significant results have been obtained only for groups not containing any elements of age 2.…”

“…By analysing the proof of Theorem 5.1 in[30] one checks that the G-Hilb resolution corresponds to the chamber σ 0 .…”

“…Flops can be performed between adjacent resolutions in rows and in columns. The G-Hilb resolution can be identified based on the proof of[30, Thm 5.2] as the central resolution on the base of the triangle.…”

Reduction of triangulated categories and maximal modification algebras for cAn singularities

“…This fact will be used in §1.5, and is also needed in the geometric setting of Nolla-Sekiya [NS,§5.5]. …”