Cyclic Quotient Singularities Research Articles

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.

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

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.

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.

Analytic torsion for log-Enriques surfaces and Borcherds product

Abstract We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type $\frac {1}{4}(1,1)$ . The moduli space of such log-Enriques surfaces with k singular points is a modular variety of orthogonal type associated with a unimodular lattice of signature $(2,10-k)$ . We prove that the invariant, viewed as a function of the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension $2$ . As a consequence, the BCOV invariant in this case is not a birational invariant, unlike the Calabi-Yau case.

Simple embeddings of rational homology balls and antiflips

Let $V$ be a regular neighborhood of a negative chain of $2$-spheres (i.e. exceptional divisor of a cyclic quotient singularity), and let $B_{p,q}$ be a rational homology ball which is smoothly embedded in $V$. Assume that the embedding is simple, i.e. the corresponding rational blow-up can be obtained by just a sequence of ordinary blow-ups from $V$. Then we show that this simple embedding comes from the semi-stable minimal model program (MMP) for $3$-dimensional complex algebraic varieties under certain mild conditions. That is, one can find all simply embedded $B_{p,q}$'s in $V$ via a finite sequence of antiflips applied to a trivial family over a disk. As applications, simple embeddings are impossible for chains of $2$-spheres with self-intersections equal to $-2$. We also show that there are (infinitely many) pairs of disjoint $B_{p,q}$'s smoothly embedded in regular neighborhoods of (almost all) negative chains of $2$-spheres. Along the way, we describe how MMP gives (infinitely many) pairs of disjoint rational homology balls $B_{p,q}$ embedded in blown-up rational homology balls $B_{n,a} # \bar{\mathbb{CP}^2}$ (via certain divisorial contractions), and in the Milnor fibers of certain cyclic quotient surface singularities. This generalizes results in [Khodorovskiy-2014], [H. Park-J. Park-D. Shin-2016], [Owens-2017] by means of a uniform point of view.

A new equivalence between singularity categories of commutative algebras

We construct a triangle equivalence between the singularity categories of two isolated cyclic quotient singularities of Krull dimensions two and three, respectively. This is the first example of a singular equivalence involving connected commutative algebras of odd and even Krull dimension. In combination with Orlov's localization result, this gives further singular equivalences between certain quasi-projective varieties of dimensions two and three, respectively.

Derived categories of singular surfaces

We develop an approach that allows one to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite-dimensional algebras. We first explain how to induce a semiorthogonal decomposition of a surface $X$ with rational singularities from a semiorthogonal decomposition of its resolution. In the case when $X$ has cyclic quotient singularities, we introduce the condition of adherence for the components of the semiorthogonal decomposition of the resolution that allows one to identify the components of the induced decomposition of $X$ with derived categories of local finite-dimensional algebras. Further, we present an obstruction in the Brauer group of $X$ to the existence of such a semiorthogonal decomposition, and show that in the presence of the obstruction a suitable modification of the adherence condition gives a semiorthogonal decomposition of the twisted derived category of $X$. We illustrate the theory by exhibiting a semiorthogonal decomposition for the untwisted or twisted derived category of any normal projective toric surface depending on whether its Weil divisor class group is torsion-free or not. For weighted projective planes we compute the generators of the components explicitly and relate our results to the results of Kawamata based on iterated extensions of reflexive sheaves of rank 1.

Nearly Gorenstein cyclic quotient singularities

We investigate the nearly Gorenstein property among d-dimensional cyclic quotient singularities Bbbk llbracket x_1,dots ,x_drrbracket ^G, where Bbbk is an algebraically closed field and Gsubseteq {text {GL}}(d,Bbbk ) is a finite small cyclic group whose order is invertible in Bbbk . We prove a necessary and sufficient condition to be nearly Gorenstein that also allows us to find several new classes of such rings.

Multidimensional continued fractions for cyclic quotient singularities and Dedekind sums

We present a multidimensional continued fraction of Hirzebruch–Jung type which controls a certain resolution of an isolated cyclic quotient singularity, and we study its geometric properties. As an application, we explicitly express a certain 3-dimensional Fourier–Dedekind sum in terms of our continued fraction.

Biregular models of log Del Pezzo surfaces with rigid singularities

We construct biregular models of families of log Del Pezzo surfaces with rigid cyclic quotient singularities such that a general member in each family is wellformed and quasismooth. Each biregular model consists of infinite series of such families of surfaces; parameterized by the natural numbers $\mathbb{N}$. Each family in these models is represented by either a codimension 3 Pfaffian format modelled on the Pl\ucker embedding of Gr(2,5) or a codimension 4 format modelled on the Segre embedding of \(\mathbb{P}^2 \times \mathbb{P}^2 \). In particular, we show the existence of two biregular models in codimension 4 which are bi parameterized, giving rise to an infinite series of models of families of log Del Pezzo surfaces. We identify those models of surfaces which do not admit a \(\mathbb {Q}\)-Gorenstein deformation to a toric variety.

The symmetric signature of cyclic quotient singularities

The symmetric signature is an invariant of local domains which was recently introduced by Brenner and the first author in an attempt to find a replacement for the $F$-signature in characteristic zero. In the present note we compute the symmetric signature for two-dimensional cyclic quotient singularities, i.e. invariant subrings $k[[u,v]]^G$ of rings of formal power series under the action of a cyclic group $G$. Equivalently, these rings arise as the completions (at the irrelevant ideal) of two-dimensional normal toric rings. We show that for this class of rings the symmetric signature coincides with the $F$-signature.

The correction term for the Riemann–Roch formula of cyclic quotient singularities and associated invariants

This paper deals with the invariant \(R_X\) called the RR-correction term, which appears in the Riemann–Roch and Numerical Adjunction Formulas for normal surface singularities. Typically, \(R_X=\delta ^\text {top}_X-\delta ^\text {an}_X\) decomposes as difference of topological and analytical local invariants of its singularities. The invariant \(\delta ^\text {top}_X\) is well understood and depends only on the dual graph of a good resolution. The purpose of this paper is to give a new interpretation for \(\delta ^\text {an}_X\), which in the case of cyclic quotient singularities can be explicitly computed via generic divisors. We also include two types of applications: one is related to the McKay decomposition of reflexive modules in terms of special reflexive modules in the context of the McKay correspondence. The other application answers two questions posed by Blache (Abh Math Semin Univ Hambg 65:307–340, 1995) on the asymptotic behavior of the invariant \(R_X\) of the pluricanonical divisor.

Calabi–Yau Double Coverings of Fano–Enriques Threefolds

AbstractThis note is a report on the observation that the Fano–Enriques threefolds with terminal cyclic quotient singularities admit Calabi–Yau threefolds as their double coverings. We calculate the invariants of those Calabi–Yau threefolds when the Picard number is one. It turns out that all of them are new examples.

Smoothing toric Fano surfaces using the Gross-Siebert algorithm

A toric del Pezzo surface $X_P$ with cyclic quotient singularities determines and is determined by a Fano polygon $P$. We construct an affine manifold with singularities that partially smooths the boundary of $P$; this a tropical version of a Q-Gorenstein partial smoothing of $X_P$. We implement a mild generalization of the Gross-Siebert reconstruction algorithm - allowing singularities that are not locally rigid - and thereby construct (a formal version of) this partial smoothing directly from the affine manifold. This has implications for mirror symmetry: roughly speaking, it implements half of the expected mirror correspondence between del Pezzo surfaces with cyclic quotient singularities and Laurent polynomials in two variables.

On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

AbstractIn earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfacesXwith isolated cyclic quotient singularities such thatXadmits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomialsfin two variables, and that the quantum period of such a surfaceX, which is a generating function for Gromov–Witten invariants ofX, coincides with the classical period of its mirror partnerf.In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with13$\begin{array}{} \frac{1}{3} \end{array} $(1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.

Lens spaces among 3-manifolds and quotient surface singularities

This is a pseudo-historical survey about some aspects of lens spaces and their relations with cyclic quotient singularities. References are ordered by the year of publication. Their list is not exhaustive.

Dual $F$-signature of special Cohen-Macaulay modules over cyclic quotient surface singularities

The notion of $F$-signature is defined by C. Huneke and G. Leuschke and this numerical invariant characterizes some singularities. This notion is extended to finitely generated modules and called dual $F$-signature. In this paper, we determine the dual $F$-signature of a certain class of Cohen-Macaulay modules (so-called "special") over cyclic quotient surface singularities. Also, we compare the dual $F$-signature of a special Cohen-Macaulay module with that of its Auslander-Reiten translation. This gives a new characterization of the Gorensteiness.

Toric geometry in polymake

We give an overview of the application 'fulton' for toric geometry of the software framework polymake. Using two-dimensional cyclic quotient singularities as our main example, we will show how to create toric varieties and divisors on these. The Singular interface of polymake allows one to go back to the algebraic side in case there is no combinatorial algorithm known yet, in order to develop new conjectures and algorithms.

On the Linear Span of Lattice Points in a Parallelepiped

Let $\Lambda\subset\mathbf{R}^{n}$ be a lattice which contains the integer lattice $\mathbf{Z}^{n}$. We characterize the space of linear functions $\mathbf{R}^{n}\rightarrow\mathbf{R}$ which vanish on the lattice points of $\Lambda$ lying in the half-open unit cube $[0,1)^{n}$. We also find an explicit formula for the dimension of the linear span of $\Lambda\cap[0,1)^{n}$. The results in this paper generalize and are based on the Terminal Lemma of Reid, which is in turn based upon earlier work of Morrison and Stevens on the classification of four dimensional isolated Gorenstein terminal cyclic quotient singularities.