Quotient Singularities Research Articles

Crepant Resolutions of Stratified Varieties via Gluing

Abstract Let $X$ be a variety with a stratification ${\mathcal{S}}$ into smooth locally closed subvarieties such that $X$ is locally a product along each stratum (e.g., a symplectic singularity). We prove that assigning to each open subset $U \subset X$ the set of isomorphism classes of locally projective crepant resolutions of $U$ defines an ${\mathcal{S}}$-constructible sheaf of sets. Thus, for each stratum $S$ and basepoint $s \in S$, the fundamental group acts on the set of germs of projective crepant resolutions at $s$, leaving invariant the germs extending to the entire stratum. Global locally projective crepant resolutions correspond to compatible such choices for all strata. For example, if the local projective crepant resolutions are unique, they automatically glue uniquely. We give criteria for a locally projective crepant resolution $\rho : \tilde X \to X$ to be globally projective. We show that the sheafification of the presheaf $U \mapsto \operatorname{Pic}(\rho ^{-1}(U)/U)$ of relative Picard classes is also constructible. The resolution is globally projective only if there exist local relatively ample bundles whose classes glue to a global section of this sheaf. The obstruction to lifting this section to a global ample line bundle is encoded by a gerbe on the singularity $X$. We show the gerbes are automatically trivial if $X$ is a symplectic quotient singularity. Our main results hold in the more general setting of partial crepant resolutions, that need not have smooth source. We apply the theory to symmetric powers and Hilbert schemes of surfaces with du Val singularities, finite quotients of tori, multiplicative, and Nakajima quiver varieties, as well as to canonical three-fold singularities.

Non-commutative resolutions of linearly reductive quotient singularities

ABSTRACT We prove the 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 2, we relate these to resolutions of singularities provided by G-Hilbert schemes and F-blowups. As an application, we establish and recover results concerning resolutions for toric singularities, as well as canonical, log terminal and F-regular singularities in dimension 2.

The class group of a minimal model of a quotient singularity

The class group of a minimal model of a quotient singularity

2 n2-inequality for cA1 points and applications to birational rigidity

The $4 n^2$ -inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type $cA_1$ , and obtain a $2 n^2$ -inequality for $cA_1$ points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree $1$ over $\mathbb {P}^1$ satisfying the $K^2$ -condition, all of which have at most terminal $cA_1$ singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a $cA_1$ point which is not an ordinary double point.

Quotient singularities in the Grothendieck ring of varieties

Let G G be a finite group, X X be a smooth complex projective variety with a faithful G G -action, and Y Y be a resolution of singularities of X / G X/G . Larsen and Lunts asked whether [ X / G ] − [ Y ] [X/G]-[Y] is divisible by [ A 1 ] [\mathbb {A}^1] in the Grothendieck ring of varieties. We show that the answer is negative if B G BG is not stably rational and affirmative if G G is abelian. The case when X = Z n X=Z^n for some smooth projective variety Z Z and G = S n G=S_n acts by permutation of the factors is of particular interest. We make progress on it by showing that [ Z n / S n ] − [ Z ⟨ n ⟩ / S n ] [Z^n/S_n]-[Z\langle n\rangle / S_n] is divisible by [ A 1 ] [\mathbb {A}^1] , where Z ⟨ n ⟩ Z\langle n\rangle is Ulyanov’s polydiagonal compactification of the n n th configuration space of Z Z .

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

We demonstrate that the linear quotient singularity for the exceptional subgroup G G in S p ( 4 , C ) \mathrm {Sp}(4,\mathbb {C}) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of C 4 / G \mathbb {C}^4/G as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the n n -pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2 n − 6 2n-6 ; for example, we show that there are 1684 projective crepant resolutions when n = 6 n=6 . We also prove that the resulting affine cones are not quotient singularities for n ≥ 6 n \geq 6 .

Fields of moduli and the arithmetic of tame quotient singularities

Given a perfect field $k$ with algebraic closure $\overline {k}$ and a variety $X$ over $\overline {k}$, the field of moduli of $X$ is the subfield of $\overline {k}$ of elements fixed by field automorphisms $\gamma \in \operatorname {Gal}(\overline {k}/k)$ such that the Galois conjugate $X_{\gamma }$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset \overline {k}$ such that $X$ descends to $k'$. In this paper, we extend the formalism and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher-dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname {Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli.

On fillings of contact links of quotient singularities

AbstractWe study several aspects of fillings for links of general isolated quotient singularities using Floer theory, including co‐fillings, Weinstein fillings, strong fillings, exact fillings and exact orbifold fillings, focusing on the non‐existence of exact fillings of contact links of isolated terminal quotient singularities. We provide an extensive list of isolated terminal quotient singularities whose contact links are not exactly fillable, including for , which settles a conjecture of Eliashberg, quotient singularities from general cyclic group actions and finite subgroups of , and all terminal quotient singularities in complex dimension 3. We also obtain uniqueness of the orbifold diffeomorphism type of exact orbifold fillings of contact links of some isolated terminal quotient singularities.

Symplectic rational blow-ups on rational 4-manifolds

We prove that if a symplectic 4-manifold X X becomes a rational 4-manifold after applying rational blow-down surgery, then the symplectic 4-manifold X X is originally rational. That is, a symplectic rational blow-up of a rational symplectic 4 4 -manifold is again rational. As an application we show that a degeneration of a family of smooth rational complex surfaces is a rational surface if the degeneration has at most quotient surface singularities, which generalizes slightly a classical result of Bădescu [J. Reine Angew. Math. 367 (1986), pp. 76–89] in algebraic geometry under a mild additional condition.

Discriminant groups of wild cyclic quotient singularities

Let p be prime. We describe explicitly the resolution of singularities of several families of wild Z/pZ-quotient singularities in dimension two, including families that generalize the quotient singularities of type E_6, E_7, and E_8 from p=2 to arbitrary characteristics. We prove that for odd primes, any power of p can appear as the determinant of the intersection matrix of a wild Z/pZ-quotient singularity. We also provide evidence towards the conjecture that in this situation one may choose the wild action to be ramified precisely at the origin.

The Arithmetic of Tame Quotient Singularities in Dimension 2

Abstract Let $k$ be a field, $X$ a variety with tame quotient singularities, and $\tilde {X}\to X$ a resolution of singularities. Any smooth rational point $x\in X(k)$ lifts to $\tilde {X}$ by the Lang–Nishimura theorem, but if $x$ is singular this might be false. For certain types of singularities, the rational point is guaranteed to lift, though; these are called singularities of type $\textrm {R}$. This concept has applications in the study of the fields of moduli of varieties and yields an enhanced version of the Lang–Nishimura theorem where the smoothness assumption is relaxed. We classify completely the tame quotient singularities of type $\textrm {R}$ in dimension $2$; in particular, we show that every non-cyclic tame quotient singularity in dimension $2$ is of type $\textrm {R}$, and most cyclic singularities are of type $\textrm {R}$ too.

On parabolic subgroups of symplectic reflection groups

AbstractUsing Cohen’s classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group, are themselves symplectic reflection groups. This is the symplectic analog of Steinberg’s Theorem for complex reflection groups.Using computational results required in the proof, we show the nonexistence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open.Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.

SINGULARITIES OF QUOTIENT SURFACES OF K3 SURFACES

We study cyclic quotient singularities given by automorphisms of maximum order on K3 surfaces. In particular, we describe fixed loci of such automorphisms, and provide types of these singularities

𝜇_{𝑝}- and 𝛼_{𝑝}-actions on K3 surfaces in characteristic 𝑝

We consider μ p \mu _p - and α p \alpha _p -actions on RDP K3 surfaces (K3 surfaces with rational double point (RDP) singularities allowed) in characteristic p > 0 p > 0 . We study possible characteristics, quotient surfaces, and quotient singularities. It turns out that these properties of μ p \mu _p - and α p \alpha _p -actions are analogous to those of Z / l Z \mathbb {Z}/l\mathbb {Z} -actions (for primes l ≠ p l \neq p ) and Z / p Z \mathbb {Z}/p\mathbb {Z} -quotients respectively. We also show that conversely an RDP K3 surface with a certain configuration of singularities admits a μ p \mu _p - or α p \alpha _p - or Z / p Z \mathbb {Z}/p\mathbb {Z} -covering by a “K3-like” surface, which is often an RDP K3 surface but not always, as in the case of the canonical coverings of Enriques surfaces in characteristic 2 2 .

Orthogonal exceptional collections from -Gorenstein degeneration of surfaces

We consider a surface that admits a -Gorenstein degeneration to a cyclic quotient singularity After assuming several technical conditions, we construct d exceptional vector bundles of rank n which are orthogonal to each other.

Cancellation for Surfaces Revisited

The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism X × A n ≅ X ′ × A n X\times \mathbb {A}^n\cong X’\times \mathbb {A}^n for (affine) algebraic varieties X X and X ′ X’ implies that X ≅ X ′ X\cong X’ . In this paper we provide a criterion for cancellation by the affine line (that is, n = 1 n=1 ) in the case where X X is a normal affine surface admitting an A 1 \mathbb {A}^1 -fibration X → B X\to B with no multiple fiber over a smooth affine curve B B . For two such surfaces X → B X\to B and X ′ → B X’\to B we give a criterion as to when the cylinders X × A 1 X\times \mathbb {A}^1 and X ′ × A 1 X’\times \mathbb {A}^1 are isomorphic over B B . The latter criterion is expressed in terms of linear equivalence of certain divisors on the Danielewski-Fieseler quotient of X X over B B . It occurs that for a smooth A 1 \mathbb {A}^1 -fibered surface X → B X\to B the cancellation by the affine line holds if and only if X → B X\to B is a line bundle, and, for a normal such X X , if and only if X → B X\to B is a cyclic quotient of a line bundle (an orbifold line bundle). If X X does not admit any A 1 \mathbb {A}^1 -fibration over an affine base then the cancellation by the affine line is known to hold for X X by a result of Bandman and Makar-Limanov. If the cancellation does not hold then X X deforms in a non-isotrivial family of A 1 \mathbb {A}^1 -fibered surfaces X λ → B X_\lambda \to B with cylinders X λ × A 1 X_\lambda \times \mathbb {A}^1 isomorphic over B B . We construct such versal deformation families and their coarse moduli spaces provided B B does not admit nonconstant invertible functions. Each of these coarse moduli spaces has infinite number of irreducible components of growing dimensions; each component is an affine variety with quotient singularities. Finally, we analyze from our viewpoint the examples of non-cancellation constructed by Danielewski, tom Dieck, Wilkens, Masuda and Miyanishi, e.a.

Higher Du Bois singularities of hypersurfaces

For a complex algebraic variety X $X$ , we introduce higher p $p$ -Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms Ω X q $\Omega _X^q$ and the shifted graded pieces of the Du Bois complex Ω ̲ X q $\underline{\Omega }_X^q$ for q ⩽ p $q\leqslant p$ . If X $X$ is a reduced hypersurface, we show that higher p $p$ -Du Bois singularity coincides with higher p $p$ -log canonical singularity, generalizing a well-known theorem for p = 0 $p=0$ . The assertion that p $p$ -log canonicity implies p $p$ -Du Bois has been proved by Mustata, Olano, Popa, and Witaszek quite recently as a corollary of two theorems asserting that the sheaves of reflexive differential forms Ω X [ q ] $\Omega _X^{[q]}$ ( q ⩽ p $q\leqslant p$ ) coincide with Ω X q $\Omega _X^q$ and Ω ̲ X q $\underline{\Omega }_X^q$ , respectively, and these are shown by calculating the depth of the latter two sheaves. We construct explicit isomorphisms between Ω X q $\Omega _X^q$ and Ω ̲ X q $\underline{\Omega }_X^q$ applying the acyclicity of a Koszul complex in a certain range. We also improve some non-vanishing assertion shown by them using mixed Hodge modules and the Tjurina subspectrum in the isolated singularity case. This is useful for instance to estimate the lower bound of the maximal root of the reduced Bernstein–Sato polynomial in the case where a quotient singularity is a hypersurface and its singular locus has codimension at most 4.

Fujiki-Oka Resolution for Three-dimensional Cyclic Quotient Singularities via Binary Trees

For three dimensional cyclic quotient singularities of type 1r(1,a,r−a−1) (resp. type 1r(1,a,r−a)), the Fujiki-Oka resolution coincides with one of crepant resolutions (resp. an economic resolution). In this paper, we will characterize binary trees which gives the Fujiki-Oka resolution for the above two series of cyclic quotient singularities.

Quotient singularities of products of two curves

Quotient singularities of products of two curves

Inversion of adjunction for quotient singularities

Inversion of adjunction for quotient singularities