Related papers: Crepant resolutions of 3-dimensional quotient sing…

Let $G\subseteq GL(n)$ be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution $X\rightarrow \mathbb{C}^n/G$, which is based just on the geometry of the…

We prove that a quotient singularity $\mathbb{C}^n/G $ by a finite subgroup $G\subset SL_n(\mathbb{C})$ has a crepant resolution only if $G $ is generated by junior elements. This is a generalization of the result of Verbitsky [V]. We also…

We investigate Cox rings of symplectic resolutions of quotients of $\mathbb{C}^{2n}$ by finite symplectic group actions. We propose a finite generating set of the Cox ring of a symplectic resolution and prove that under a condition…

We develop a method of finding a Cox ring of a crepant resolution of a quotient singularity with a torus action and apply it to examples of symplectic quotient singularities in dimension 4. In addition we obtain a bound on the degrees of…

Let G be a finite subgroup of SL(n,C), then the quotient C^n/G has a Gorenstein canonical singularity. Bridgeland-King-Reid proved that the G-Hilbert scheme Hilb^G(C^3) gives a crepant resolution of the quotient C^3/G for any finite…

The purpose of this paper is to construct a crepant resolution of quotient singularities by finite subgroups of SL(3,C) of monomial type, and prove that the Euler number of the resolution is equal to the number of conjugacy classes. This…

We construct and enumerate all crepant resolutions of hyperpolygon spaces, a family of conical symplectic singularities arising as Nakajima quiver varieties associated to a star-shaped quiver. We provide an explicit presentation of the Cox…

For which finite subgroups G of SL(r,C), r \geq 4, are there crepant desingularizations of the quotient space C^r/G? A complete answer to this question (also known as "Existence Problem" for such desingularizations) would classify all those…

We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The…

The purpose of this paper is to construct a crepant resolution of quotient singularities by trihedral groups ( finite subgroups of SL(3,C) of certain type ), and prove that each Euler number of the minimal model is equal to the number of…

In this paper, we show a condition for two-parameter Gorenstein cyclic quotient singularities to have a crepant resolution by using the remainder polynomial in any dimension.

We prove existence of non-commutative crepant resolutions (in the sense of van den Bergh) of quotient singularities by finite and linearly reductive group schemes in positive characteristic. In dimension two, we relate these to resolutions…

Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of…

This is the final draft, containing very minor proof-reading corrections. Let G in SL(n,\C) be a finite subgroup and \fie: Y -> X = \C^n/G any resolution of singularities of the quotient space. We prove that crepant exceptional prime…

We discuss some "folklore" results on categorical crepant resolutions for varieties with quotient singularities.

Let $V$ be a complex vector space on which a finite group $G$ acts by linear transformations. Let $W = V \oplus V^*$ be the sum of $V$ with its dual $V^*$. We prove that if the quotient $W/G$ admits a smooth crepant resolution, then the…

An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces $\Bbb{C}^{r}/G$ in dimensions $r\geq 4$ would primarily demand the existence of projective, crepant, full desingularizations. Since this is not…

We study generators and relations of Cox rings of K3 surfaces of Picard number two. In particular we consider the Cox rings of classical examples of K3 surfaces, such as quartic surfaces containing a line and elliptic K3 surfaces.

We study Cox rings of normal threefolds on which SL2 acts with a dense orbit. Exploiting the method of U-invariants, we obtain combinatorial criteria for the total coordinate space and the base variety to have log terminal singularities.…

The McKay correspondence has had much success in studying resolutions of 3-fold quotient singularities through a wide range of tools coming from geometry, combinatorics, and representation theory. We develop a computational perspective in…