On Hypersurface Quotient Singularity of Dimension 4
Li Chiang, Shi-Shyr Roan
备注：
27 pages, Latex, 11 figures, Some reorganizations and improvement of presentations, Typos corrected, Arguments of Theorem 1 of section 3 in the earlier version are refined with clearer explanation for the justification of contradicting statement appeared in a published journal literature by some other author
We consider geometrical problems on Gorenstein hypersurface orbifolds of dimension $n \geq 4$ through the theory of Hilbert scheme of group orbits. For a linear special group $G$ acting on $\CZ^n$, we study the $G$-Hilbert scheme, $\hl^G(\CZ^n)$, and crepant resolutions of $\CZ^n/G$ for $G$=the $A$-type abelian group $ A_r(n)$. For $n=4$, we obtain the explicit structure of $\hl^{A_r(4)}(\CZ^4)$. The crepant resolutions of $\CZ^4/A_r(4)$ are constructed through their relation with $\hl^{A_r(4)}(\CZ^4)$, and the connections between these crepant resolutions are found by the "flop" procedure of 4-folds. We also make some primitive discussion on $\hl^G(\CZ^n)$ for the $G$= alternating group ${\goth A}_{n+1}$ of degree $n+1$ with the standard representation on $\CZ^n$; the detailed structure of $\hl^{{\goth A}_4}(\CZ^3)$ is explicitly constructed.
We study the intersection of two particular Fermat hypersurfaces in $\mathbb{P}^3$ over a finite field. Using the Kani-Rosen decomposition we study arithmetic properties of this curve in terms of its quotients. Explicit computation of the quotients is done using a Gröbner basis algorithm. We also study the $p$-rank, zeta function, and number of rational points, of the modulo $p$ reduction of the curve. We show that the Jacobian of the genus 2 quotient is $(4,4)$-split.
We suggest to endow Mumford's GIT quotient scheme with a stack structure, by replacing Proj(-) of the invariant ring with its stack theoretic analogue. We analyse the stacks resulting in this way from classically studied invariant rings, and in particular for binary forms of low degree. Our viewpoint is that the stack structure carries interesting geometric information that is intrinsically present in the invariant ring, but lost when passing to its Proj(-).
In this paper, we will show that the Hesselink stratification of a Hilbert scheme of hypersurfaces is independent of the choice of Plücker coordinate and there is a positive relation between the length of Hesselink's worst virtual 1-parameter subgroup and multiplicity of a projective hypersurface.
In [1] some quotients of one-parameter families of Calabi-Yau varieties are related to the family of Mirror Quintics by using a construction due to Shioda. In this paper, we generalize this construction to a wider class of varieties. More specifically, let $A$ be an invertible matrix with non-negative integer entries. We introduce varieties $X_A$ and $\overline{M}_A$ in weighted projective space and in ${\mathbb P}^n$, respectively. The variety $\overline{M}_A$ turns out to be a quotient of a Fermat variety by a finite group. As a by-product, $X_A$ is a quotient of a Fermat variety and $\overl…
Let $(X,0)$ be an isolated hypersurface singularity defined by $ϕ\colon(\mathbb C^n,0)\to(\mathbb C,0)$ and $f\colon(\mathbb C^n,0)\to\mathbb C$ such that the Bruce-Roberts number $μ_{BR}(f,X)$ is finite. We first prove that $μ_{BR}(f,X)=μ(f)+μ(ϕ,f)+μ(X,0)-τ(X,0)$, where $μ$ and $τ$ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen-Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ …
The set of volumes of stable surfaces does have accumulation points. In this paper, we study this phenomenon for surfaces with one cyclic quotient singularity, towards answering the question under which conditions we can still have boundedness. Effective bounds allow listing singularities that might appear on a stable surface after fixing its invariants. We find optimal inequalities for stable surfaces with one cyclic quotient singularity, which can be used to prove boundedness under certain conditions. We also introduce the notion of generalized T-singularity, which is a natural generalizatio…
We show that a homology plane of general type has at worst a single cyclic quotient singular point. An example of such a surface with a singular point does exist. We also show that the automorphism group of a smooth contractible surface of general type is cyclic.
We introduce two generalizations of Newton-non-degenerate (Nnd) singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called topologically Newton-non-degenerate (tNnd) if the local embedded topological singularity type can be restored from a collection of Newton diagrams (for some coordinate choices). A singularity that is not tNnd is called essentially Newton-degenerate. For plane curves we give an explicit characterization of tNnd singularities; for hypersurfaces we provide several examples. Next, we treat the question: whether Nnd or tNnd is a property of…
We build two embedded resolution procedures of a quasi-ordinary singularity of complex analytic hypersurface, by using toric morphisms which depend only on the characteristic monomials associated to a quasi-ordinary projection of the singularity. This result answers an open problem of Lipman in Equisingularity and simultaneous resolution of singularities, Resolution of Singularities, Progress in Mathematics No. 181, 2000, 485-503. In the first procedure the singularity is embedded as hypersurface. In the second procedure, which is inspired by a work of Goldin and Teissier for plane curves (see…
In this note we give a simple proof of the following relative analog of the well known Milnor-Palamodov theorem: the Bruce-Roberts number of a function relative to an isolated hypersurface singularity is equal to its topological Milnor number (the rank of a certain relative (co)homology group) if and only if the hypersurface singularity is quasihomogeneous. The proof relies on an interpretation of the Bruce-Roberts number in terms of differential forms and the Lê-Greuel formula.
A version of Kontsevich Formality theorem is proven for smooth DG algebras. As an application of this, it is proven that any quasiclassical datum of noncommutative unfolding of an isolated surface singularity can be quantized.
The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a $\Q$-Gorenstein singularity.
This paper studies hypersurface exceptional singularities in $\mathbb C^n$ defined by non-degenerate function. For each canonical hypersurface singularity, there exists a weighted homogeneous singularity such that the former is exceptional if and only if the latter is exceptional. So we study the weighted homogeneous case and prove that the number of weights of weighted homogeneous exceptional singularities are finite. Then we determine all exceptional singularities of the Brieskorn type of dimension 3.
It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type o…
In this paper we give a complete answer to a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of a plane curve singularity. We put this question into a general framework of the study of the difference of Milnor and Tjurina numbers for isolated complete intersection singularities showing its connection with other problems in singularity theory.
For a weighted quasihomogeneous two dimensional hypersurface singularity, we define a smoothing with unipotent monodromy and an isolated graded normal singularity. We study the natural weighted blow up of both the smoothing and the surface. In particular, we describe our construction for the quasihomogeneous singularities of type I, the 14 unimodal exceptional singularities and we relate it to their stable replacement.
A relation is proved between the Poincaré series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. For a Kleinian singularity not of type $A_{2n}$, this amounts to the statement that the Poincaré series is the quotient of the characteristic polynomial of the Coxeter element by the characteristic polynomial of the affine Coxeter element of the corresponding root system. We show that this result also follows from the McKay correspondence.
We prove that certain roots of the Bernstein-Sato polynomial (i.e. b-function) are jumping coefficients up to a sign, showing a partial converse of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith, and D. Varolin. We also prove that certain roots are determined by a filtration on the Milnor cohomology, generalizing a theorem of B. Malgrange in the isolated singularity case. This implies a certain relation with the spectrum which is determined by the Hodge filtration, because the above filtration is related to the pole order filtration. For multiplier ideals we prove an explicit formula in the ca…
Given a nonconstant polynomial map over the reals having an isolated critical point in the origin and with zero locus of positive dimension, we establish a formula for the singular homology groups of a Milnor fibre relative to its boundary.
A topological interpretation of Hochster's Theta pairing of two modules on a hypersurface ring is given in terms of linking numbers. This generalizes results of M. Hochster and proves a conjecture of J. Steenbrink. As a corollary we get that the Theta pairing vanishes for isolated hypersurface singularities in an odd number of variables, as was conjectured by H. Dao.
In this paper the singular hypersurfaces in $\mathbb{C}\mathrm{P}^4$ of degree $d$ with an isolated singularity are studied. If the singularity is of type $A_{2k+1}$, under the condition $d<(k+5)/2$, a classification of such hypersurfaces upto homeomorphism (which is diffeomorphism on the nonsingular part) is obtained. As an application, cubic hypersurfaces with an $A_5$-singularity are constructed.
Let a finite abelian group $G$ act (linearly) on the space $\mathbb{R}^n$ and thus on its complexification $\mathbb{C}^n$. Let $W$ be the real part of the quotient $\mathbb{C}^n/G$ (in general $W \neq \mathbb{R}^n/G$). We give an algebraic formula for the radial index of a 1-form on the real quotient $W$. It is shown that this index is equal to the signature of the restriction of the residue pairing to the $G$-invariant part $Ω^G_ω$ of $Ω_ω= Ω^n_{\mathbb{R}^n,0}/ω\wedge Ω^{n-1}_{\mathbb{R}^n,0}$. For a $G$-invariant function $f$, one has the so-called quantum cohomology group defined in the qu…
We associate to any irreducible germ S of complex quasi-ordinary hypersurface an analytically invariant semigroup. We deduce a direct proof (without passing through their embedded topological invariance) of the analytical invariance of the normalized characteristic exponents. These exponents generalize the generic Newton-Puiseux exponents of plane curves. Incidentally, we give a toric description of the normalization morphism of the germ S.
We study the GIT compactifications of pairs formed by a hypersurface and a hyperplane. We provide a general setting to characterize all polarizations which give rise to different GIT quotients. Furthermore, we describe a finite set of one-parameter subgroups sufficient to determine the stability of any GIT quotient. We characterize all maximal orbits of non stable and strictly semistable pairs, as well as minimal closed orbits of strictly semistable pairs. Our construction gives natural compactifications of the space of log smooth pairs for Fano and Calabi-Yau hypersurfaces.
For a complex algebraic variety $X$, we introduce higher $p$-Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms $Ω_X^q$ and the shifted graded pieces of the Du Bois complex $\underlineΩ_X^q$ for $q\le p$. If $X$ is a reduced hypersurface, we show that higher $p$-Du~Bois singularity coincides with higher $p$-log canonical singularity, generalizing a well-known theorem for $p=0$. The assertion that $p$-log canonicity implies $p$-Du Bois has been proved by Mustata, Olano, Popa, and Witaszek quite recently as a corollary of two theorems assertin…
We enumerate complex algebraic hypersurfaces in $P^n$, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equi-singular strata in the parameter space of hypersurfaces. We suggest an inductive procedure, based on intersection theory combined with liftings and degenerations. The procedure computes the (co)homology class in question, whenever a given singularity type is properly defined and the stratum possesses good geometric properties. We consider in details the generalized Newton-non-degenerate…
For a finite subgroup $G$ of $\operatorname{GL}(2, \mathbb C)$, we consider the moduli space ${\mathcal M}_θ$ of $G$-constellations. It depends on the stability parameter $θ$ and if $θ$ is generic it is a resolution of singularities of $\mathbb C^2/G$. In this paper, we show that a resolution $Y$ of $\mathbb C^2/G$ is isomorphic to ${\mathcal M}_θ$ for some generic $θ$ if and only if $Y$ is dominated by the maximal resolution under the assumption that $G$ is abelian or small.
The Aluffi algebra is algebraic definition of characteristic cycles of a hypersurface in intersection theory. In this paper we focus on the Aluffi algebra of quasi-homogeneous and locally Eulerian hypersurface with isolated singularities. We prove that the Jacobian ideal of an affine hypersurfac with isolated singularities is of linear type if and only if it is locally Eulerian. We show that the gradient ideal of a projective hypersurface is of linear type if and only if the corresponding affine curve in the affine chart associated to singular points is locally Eulerian. We prove that the grad…
Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）；Escuela de Ciencias Físicas y Matemáticas, Universidad de San Carlos de Guatemala（危地马拉圣卡洛斯大学物理与数学院）；Aix Marseille Univ, CNRS, I2M（艾克斯-马赛大学，法国国家科学研究中心，I2M）
Let Y be a hypersurface in projective space having only ordinary double points as singularities. We prove a variant of a conjecture of L. Wotzlaw on an algebraic description of the graded quotients of the Hodge filtration on the top cohomology of the complement of Y except for certain degrees of the graded quotients, as well as its extension to the Milnor cohomology of a defining polynomial of Y for degrees a little bit lower than the middle. These partially generalize theorems of Griffiths and Steenbrink in the Y smooth case, and enable us to determine the structure of the pole order spectral…
We compute explicitly the limits of tangents of a quasi-ordinary singularity in terms of its special monomials. We show that the set of limits of tangents of Y is essentially a topological invariant of Y .
In this paper we extend the concept of Milnor fiber and Milnor number of a curve singularity allowing the ambient space to be a quotient surface singularity. A generalization of the local δ-invariant is defined and described in terms of a Q-resolution of the curve singularity. In particular, when applied to the classical case (the ambient space is a smooth surface) one obtains a formula for the classical δ-invariant in terms of a Q-resolution, which simplifies considerably effective computations. All these tools will finally allow for an explicit description of the genus formula of a curve def…
We provide several results on splice-quotient singularities: a combinatorial expression of the dimension of the first cohomology of all `natural' line bundles, an equivariant Campillo-Delgado-Gusein-Zade type formula about the dimension of relative sections of line bundles (proving that the equivariant, divisorial multi-variable Hilbert series is topological), a combinatorial description of divisors of analytic function-germs, and an expression for the multiplicity of the singularity from its resolution graph. Additional, we establish a new formula for the Seiberg-Witten invariants of any rati…
We investigate some coboundary map associated to a $3$-dimensional terminal singularity which is important in the study of deformations of singular $3$-folds. We prove that this map vanishes only for quotient singularities and a $A_{1,2}/4$-singularity, that is, a terminal singularity analytically isomorphic to a $\mathbb{Z}_4$-quotient of the singularity $ (x^2+y^2 +z^3+u^2=0)$. As an application, we prove that a $\mathbb{Q}$-Fano $3$-fold with terminal singularities can be deformed to one with only quotient singularities and $A_{1,2}/4$-singularities. We also treat the $\mathbb{Q}$-smoothabi…
We show that Braun-Chuang-Lazarev's derived quotient prorepresents a naturally defined noncommutative derived deformation functor. Given a noncommutative partial resolution of a Gorenstein algebra, we show that the associated derived quotient controls part of its dg singularity category. We use a recent result of Hua and Keller to prove a recovery theorem, which can be thought of as providing a solution to a derived enhancement of a conjecture made by Donovan and Wemyss about the birational geometry of threefold flops.
We determine a one-to-one correspondence between Milnor fibers and minimal symplectic fillings of a quotient surface singularity (up to diffeomorphism type) by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3-folds and Pinkham's negative weight smoothing. As by-products, we show that: -- Milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions. For this, we classify minimal symplectic fillings of a quotient …
Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence of the map from $V_{n,d}$ to the homotopy quotient $(V_{n,d})_{hG'}$ degenerates at $E_2$, and so does the Leray spectral sequence of the quotient map $V_{n,d}\to V_{n,d}/G'$ provided the geometric quotient $V…
We show that for every morphism f between nonsingular hypersurfaces of dimension at least 3 and of general type in projective space, there is an everywhere defined endomorphism F of projective space that restricts to f. As a corollary, we see that if X,Y are nonsingular hypersurfaces of general type of dimension at least 3 such that there is a nonconstant morphism f from X to Y, then degY divides degX with quotient q, and moreover the endomorphism F of projective space is given by polynomials of degree q.
In this paper, we prove that the Milnor fibre of a singularity over an i.c.i.s. of dimension 3 has the homotopy type of a bouquet of spheres, provided that the function that defines the singularity has finite extended codimension with respect to the ideal that defines the i.c.i.s.
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 second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.
We show that a cyclic quotient surface singularity S can be decomposed, in a precise sense, into a number of elementary T-singularities together with a cyclic quotient surface singularity called the residue of S. A normal surface X with isolated cyclic quotient singularities {S_i} admits a Q-Gorenstein partial smoothing to a surface with singularities given by the residues of the S_i. We define the singularity content of a Fano lattice polygon P: this records the total number of elementary T-singularities and the residues of the corresponding toric Fano surface X_P. We express the degree of X_…
Given a prime $\ell\geq 3$ and a positive integer $k \le \ell-2$, one can define a matrix $D_{k,\ell}$, the so-called Demjanenko matrix, whose rank is equal to the dimension of the Hodge group of the Jacobian ${\mathrm Jac}({\mathcal C}_{k,\ell})$ of a certain quotient of the Fermat curve of exponent $\ell$. For a fixed $\ell$, the existence of $k$ for which $D_{k,\ell}$ is singular (equivalently, for which the rank of the Hodge group of ${\mathrm Jac}({\mathcal C}_{k,\ell})$ is not maximal) has been extensively studied in the literature. We provide an asymptotic formula for the number of such…
We give a generalization of an algebraic formula of Gomez-Mont for the index of a vector field with isolated zero in (C^n,0) and tangent to an isolated hypersurface singularity. We only assume that the vector field has an isolated zero on the singularity.
Using his deep and beautiful idea of cutting with a Hyperplane, Lefschetz explained how the homology groups of a projective smooth variety could be constructed from basic pieces, that he called primitive homology. This idea can be applied every time we have a vector space with a biliner form (i.e. homology with cup product) and a $\pm$-symmetric nilpotent operator (i.e. cutting with a hyperplane). We will illustrate this in the context of Singularity Theory: A germ of an isolated singular point of a hypersurface defined by $f=0$. We begin with the algebraic setting in the Jacobian (or Milnor) …
