Related papers: Computing resolutions of quotient singularities

We study Cox rings of crepant resolutions of quotient singularities $\mathbb{C}^3/G$ where $G$ is a finite subgroup of $SL(3,\mathbb{C})$. We use them to obtain information on the geometric structure of these resolutions, number of…

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 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 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…

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…

We introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining conventional resolutions using coordinate rings of matrix-valued functions. We verify this…

Let $G$ be a finite group of order $n$, and $\xi$ an $n$-th primitive root of unity. Consider the affine scheme $C:=\mbox{Spc}({\mathbb Z}[\xi]\otimes_{\mathbb Z} R(G))$ where $R(G)$ is the representation ring of $G$. We study the fibers of…

In this note we introduce Cox rings of singularities and explicitly compute them in the case of du Val singularities $\mathbb{D}_n,\mathbb{E}_6,\mathbb{E}_7$ and $\mathbb{E}_8$.

We define the equivariant Cox ring of a normal variety with algebraic group action. We study algebraic and geometric aspects of this object and show how it is related to the ordinary Cox ring. Then, we specialize to the case of normal…

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.…

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…

In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely…

We present a classification algorithm for isolated hypersurface singularities of corank 2 and modality 1 over the real numbers. For a singularity given by a polynomial over the rationals, the algorithm determines its right equivalence class…

Given a klt singularity $(X,\Delta;x)$, we define the iteration of Cox rings of $(X,\Delta;x)$. The first result of this article is that the iteration of Cox rings ${\rm Cox}^{(k)}(X,\Delta;x)$ of a klt singularity stabilizes for $k$ large…

For a variety with a finitely generated total coordinate ring, we describe basic geometric properties in terms of certain combinatorial structures living in its divisor class group. For example, we describe the singularities, we calculate…

Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring K[X]^G in the case where G is reductive. Furthermore, we address the case where G is connected and…

We present a new eigenvalue method for solving a system of Laurent polynomial equations defining a zero-dimensional reduced subscheme of a toric compactification $X$ of $(\mathbb{C} \setminus \{0\})^n$. We homogenize the input equations to…

We study Cox rings of K3-surfaces. A first result is that a K3-surface has a finitely generated Cox ring if and only if its effective cone is polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of…

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…

Let G be a connected reductive group and G/H a spherical homogeneous space. We show that the ideal of relations between a natural set of generators of the Cox ring of a G-embedding of G/H can be obtained by homogenizing certain equations…