ON SMOOTHNESS OF MINIMAL MODELS OF QUOTIENT SINGULARITIES BY FINITE SUBGROUPS OF SL_n(ℂ)

Abstract

We prove that a quotient singularity C n /G by a finite subgroup G ⊂ SL n (C) has a crepant resolution only if G is generated by junior elements. This is a generalization of the result of Verbitsky [V]. We also give a procedure to compute the Cox ring of a minimal model of a given C n /G explicitly from information of G. As an application, we investigate the smoothness of minimal models of some quotient singularities. Together with work of Bellamy and Schedler, this completes the classification of symplectically imprimitive quotient singularities which admit projective symplectic resolutions.

Crepant Resolutions of 3-Dimensional Quotient Singularities via Cox Rings

“…It was introduced in the case of symplectic singularities and resolutions in [17,14]. Then, in [33] it was noticed that the same is true, and the proofs work in the same way, in the case of any minimal model of a quotient singularity for G ⊂ SL(n, C), in particular a crepant resolution if it exists. We start from embedding the Cox ring in a bigger ring which is easier to understand.…”

“…For irreducible representations we develop a different approach, because the minimal generating set does not usually give a generating set of R(X) -it needs at least linear modifications to increase valuations, but sometimes also adding more generators. We verify that obtained elements generate R(X) by applying procedures in the library developed for [15] or by the algorithm from [33].…”

“…In the notation therein, F 0 = min j (F ), p i,0 = min j (p i ) and min j (I) = (R 0 ) = min j (J). One may also check that all steps of the algorithm in [33,Sect. 4] end without introducing additional generators which gives a different proof of Theorem 4.4.…”

“…Finally, one has to verify that the ring generated by the obtained set is really a Cox ring. This can be done either with the Singular [12] library quotsingcox.lib [15] accompanying [16] or with the algorithm for finding the Cox ring of minimal models of quotient singularities from [33]. In each of presented cases we try both methods and at least one works (that is, computations end in a reasonable amount of time).…”

“…In each of presented cases we try both methods and at least one works (that is, computations end in a reasonable amount of time). Note that, however, the algorithm from [33] does not behave very well in the case when the candidate for the generating set of the Cox ring is not correct, hence we use it only for verification, not for determining elements of the generating set.…”