Resolution graphs of some surface singularities II. (generalized Iomdin series)
András Némethi, Ágnes Szilárd
备注：
36 pages, AMS-Latex, 50 figures, uses xr.sty (included), to appear in "Proceedings of the conference on singularities in algebraic and analyitic geometry" (conm, ams)
In this article we give a construction of the resolution graphs of hypersurface surface singularities (X_k,0) given by generalized Iomdin series. All these resolution graphs are coordinated by an ``universal bi-colored graph'' which is associated with the ICIS determining the Iomdin series. The definition of this new graph is rather involved, and in concrete examples it is difficult to compute. Nevertheless, we present a large number of examples. This is very helpful in the exemplification of its properties as well. We then present a construction of the resolution graphs of the surface singularities above which uses the "universal bi-coloured graph" and the integer k. This is formulated in a purely combinatorial algorithm. The result is a highly non-trivial generalization of the case of cyclic coverings of smooth surfaces.
Our purpose is to investigate all defined Poincaré series associated with multi-index filtrations and value semigroups of curve singularities---not necessarily complex---with regard to the property of forgetting variables, i.e., by making variables of the series to be 1. Generalised Poincaré series of motivic nature will be also considered.
We unify and generalize formulas obtained by Campillo, Delgado and Gusein-Zade in their series of articles. Positive results are established for rational and minimally elliptic singularities. By examples and counterexamples we also try to find the `limits' of these identities. Connections with the Seiberg-Witten Invariant Conjecture and Semigroup Density Conjecture are discussed.
We consider Fuchsian singularities of arbitrary genus and prove, in a conceptual manner, a formula for their Poincaré series. This uses Coxeter elements involving Eichler-Siegel transformations. We give geometrical interpretations for the lattices and isometries involved, lifting them to triangulated categories.
A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V. I. Arnold and V. I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities. We show that the characteristic polynomial of the monodromy is related to the Poincaré series of the coordinate algebra of the ambient singularity.
Lejeune-Jalabert and Reguera computed the geometric Poincare series P_{geom}(T) for toric surface singularities. They raise the question whether this series equals the arithmetic Poincare series. We prove this equality for a class of toric varieties including the surfaces, and construct a counterexample in the general case. We also compute the motivic Igusa Poincare series Q_{geom}(T) for toric surface singularities, using the change of variables formula for motivic integrals, thus answering a second question of Lejeune-Jalabert and Reguera's. The series Q_{geom}(T) contains more information t…
We target multivariable series associated with resolutions of complex analytic normal surface singularities. In general, the equivariant multivariable analytical and topological Poincaré series are well-defined and have good properties only if the link is a rational homology sphere. We wish to create a model when this assumption is not valid: we analyse the case of cusps. For such germs we define even the motivic versions of these two series, we prove that they are equal, and we provide explicit combinatorial expression for them. This is done via a motivic multivariable series associated with …
We give a conceptual proof that the Poincaré series of the coordinate algebra of a Kleinian singularity and of a Fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two Coxeter elements. These Coxeter elements are interpreted geometrically, using triangulated categories and spherical twist functors.
We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme $\mathcal{M}(k), ~ k \geq 3$ of coherent semistable rank 2 sheaves with Chern classes $c_1=0,~ c_2=k,~ c_3=0$ on $\mathbb{P}^3$ whose general points are sheaves with singularities of mixed dimension. These sheaves are constructed by elementary transformations of stable and properly $μ$-semistable reflexive sheaves along disjoint union of collections of points and smooth irreducible curves which are rational or complete intersection curves. As a special member of this series we obtain a new compo…
In previous papers, there were computed the Poincare series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincare series were written as the integer parts of certain fractional power series, an interpretation of whom was not given. Here we show that, up to a simple change of variables, these fractional power series are specializations of the equivariant Poincare series for filtrations on the ring of germs of functions on the universal abelian cover of the surface singularity. We compute these equivariant Poincare series.
The geometric motivic Poincaré series of a germ $(S,0)$ of complex algebraic variety takes into account the classes in the Grothendieck ring of the jets of arcs through $(S,0)$. Denef and Loeser proved that this series has a rational form. We give an explicit description of this invariant when $(S,0)$ is an irreducible germ of quasi-ordinary hypersurface singularity in terms of the Newton polyhedra of the logarithmic jacobian ideals. These ideals are determined by the characteristic monomials of a quasi-ordinary branch parametrizing $(S,0)$.
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 always possible, it is natural to ask about special classes of such quotient spaces which would satisfy the above property. In this paper we give explicit necessary and sufficient conditions under which 2-parameter series of Gorenstein cyclic quotient singularities have torus-equivariant resolutions of this specific sort in all dimensions.
The geometric motivic Poincaré series of a variety, which was introduced by Denef and Loeser, takes into account the classes in the Grothendieck ring of the sequence of jets of arcs in the variety. Denef and Loeser proved that this series has a rational form. We describe it in the case of an affine toric variety of arbitrary dimension. The result, which provides an explicit set of candidate poles, is expressed in terms of the sequence of Newton polyhedra of certain monomial ideals, which we call logarithmic jacobian ideals, associated to the modules of differential forms with logarithmic poles…
In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincaré series of this filtration were computed for simple surface singularities in the 3-space. Here they are computed for surface singularities from Arnold's lists including uni- and bimodular ones. The classification of the unimodular singularities by these Poincaré series turns out to be in accordance with their hierarchy defined by E. Brieskorn using the adjacency r…
In this manuscript we investigate the analouge of the Brill-Noether problem for smooth curves in the case of normal surface singularities. We determine the maximal possible value of $h^1$ of line bundles without fixed components in the Picard group $\pic^{l'}(\tX)$ in the following cases: for some special Chern classes $l'$ if $\tX$ is a resolution of a splice quotient singularity $(X, 0)$ and for arbitrary Chern classes in the case of weighted homogenous singularities. Motivated by this problem, we define the \emph{virtual cohomology numbers} $h^1_{virt}(l')$ for all Chern classes $l'$ such t…
For a certain class of varieties X, we derive a formula for the valuation d_{X} on the arc space L(Y) of a smooth ambient space Y, in terms of an embedded resolution of singularities. A simple transformation rule yields a formula for the geometric Poincare series. Furthermore, we prove that for this class of varieties, the arithmetic and the geometric Poincare series coincide. We also study the geometric valuation for plane curves.
In this article we define a Poincare series on a subspace of a complex analytic germ, induced by a multi-index filtration on the ambient space. We compute this Poincare series for subspaces defined by principal ideals. For plane curve singularities and nondegenerate singularities this Poincare series yields topological and geometric information. We compare this Poincare series with the one introduced in [E,G-Z]. In few cases they are equal and we show that the Poincare series we consider in this paper in general yields more information.
We study properties of the Hirzebruch class of quotient singularities $\mathbb{C}^n/G$, where $G$ is a finite matrix group. The main result states that the Hirzebruch class coincides with the Molien series of $G$ under suitable substitution of variables. The Hirzebruch class of a crepant resolution can be described specializing the orbifold elliptic genus constructed by Borisov and Libgober. It is equal to the combination of Molien series of centralizers of elements of $G$. This is an incarnation of the McKay correspondence. The results are illustrated with several examples, in particular of 4…
Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this paper we develop a general method for calculating Thom polynomials of contact singularities. Along the way, relations with the equivariant geometry of (punctual, local) Hilbert schemes, and with iterate…
The Hilbert series of a polarised algebraic variety $(X,D)$ is a powerful invariant that, while it captures some features of the geometry of $(X,D)$ precisely, often cannot recover much information about its singular locus. This work explores the extent to which the Hilbert series of an orbifold del Pezzo surface fails to pin down its singular locus, which provides nonexistence results describing when there are no orbifold del Pezzo surfaces with a given Hilbert series, supplies bounds on the number of singularities on such surfaces, and has applications to the combinatorics of lattice polytop…
We show that the generating series of Euler characteristics of Hilbert schemes of points on any algebraic surface with at worst $A_n$-type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the Dedekind eta function. In particular it is a Fourier development of a meromorphic modular form with possibly half integer weight. The key ingredient is to apply the flop transformation formula of Donaldson-Thomas type invariants counting two dimensional torsion sheaves on 3-folds proved in the author's previous paper.
In this note we investigate the generating series of the Euler characteristics of Hilbert scheme of points on cyclic quotient singularities of type (p,1). We link the appearing combinatorics to p-fountains, a generalization of the notion of fountain of coins. We obtain a representation of the generating series as coefficient of a two variable generating series.
This paper is a sequel to [He11] and [GH17]. In [He11] a notion of marking of isolated hypersurface singularities was defined, and a moduli space $M_μ^{mar}$ for marked singularities in one $μ$-homotopy class of isolated hypersurface singularities was established. It is an analogue of a Teichmüller space. It comes together with a $μ$-constant monodromy group $G^{mar}\subset G_{\mathbb{Z}}$. Here $G_{\mathbb{Z}}$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_μ^{mar}$ is connected. This is equivalent to $G^{mar}= G_{\mathbb{Z}}$. Als…
We show that the number of generators of the n-th cotangent cohomology group (n >=2) is the same for all rational surface singularities Y. For a large class of rational surface singularities, including quotient singularities, this number is also the dimension. For them we obtain an explicit formula for the corresponding Poincare series.
We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity reappears after the last blowup, with just certain exponents of the defining power series shifted upwards. The construction hence yields a cycle. Iterating this cycle leads to an infinite increase of the residual order of the defining power series. This disproves a theorem claimed by Moh about the stability of the residual order under sequences of blowups. It i…
We study the multiplicity of the jumping numbers of an $\mathfrak m$-primary ideal $\mathfrak a$ in a two-dimensional local ring with a rational singularity. The formula we provide for the multiplicities leads to a very simple and efficient method to detect whether a given rational number is a jumping number. We also give an explicit description of the Poincaré series of multiplier ideals associated to $\mathfrak a$ proving, in particular, that it is a rational function.
This is Part II of the series of our papers under the title "Toward resolution of singularities over a field of positive characteristic (The Idealistic Filtration Program)". See http://arxiv.org/abs/math/0607009 for Part I.
In the author's paper ''Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity'' 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. We study this relation for Fuchsian singularities and show that it is connected with the mirror symmetry of K3 surfaces and with automorphisms of the Leech lattice. We also indicate relations between other singularities and Conway's group.
In this paper we prove that the Gorenstein cyclic quotient singularities of type \frac 1l (1,..., 1,l-(r-1)) with $l\geq r\geq 2$, have a \textit{unique}torus-equivariant projective, crepant, partial resolution, which is ``full'' iff either $l\equiv 0$ mod $% (r-1) $ or $l\equiv 1$ mod $(r-1) $. As it turns out, if one of these two conditions is fulfilled, then the exceptional locus of the full desingularization consists of $\lfloor \frac{l}{r-1} \rfloor$ prime divisors, $\lfloor \frac{l}{r-1}\rfloor - 1$ of which are isomorphic to the total spaces of $\Bbb{P}_{\Bbb{C}}^1$-bundles over $\Bbb{P…
This text is the write-up of a series of lectures on the asymptotic mixed Hodge theory of isolated hypersurface singularities, held at the Third Latin American school on Algebraic Geometry and its applications (ELGA 3) in Guanajuato, Mexico, in august 2017. Its focus is on the classical application of the semi-continuity of the spectrum due to Varchenko and Steenbrink to the problem of bounding the possible singularities on a projective hypersurface.
Let $f(Y)\in K[[X_1,\dots,X_d]][Y]$ be a quasi-ordinary Weierstrass polynomial with coefficients in the ring of formal power series over an algebraically closed field of characteristic zero. In this paper we study the discriminant $D_f$ of $f(Y)-V$, where $V$ is a new variable. We show that the Newton polytope of $D_f$ depends only on contacts between the roots of $f(Y)$. Then we prove that $f(Y)$ is irreducible if and only if the Newton polytope of $D_f$ satisfies some arithmetic conditions. Finally we generalize these results to quasi-ordinary power series.
Kollár gave a series of examples of rational surfaces of Picard number $1$ with ample canonical divisor having cyclic singularities. In this paper, we construct several series of new examples in a geometric way, i.e., by blowing up several times inside a configuration of curves on the projective plane and then by contracting chains of rational curves. One series of our examples have the same singularities as Kollár's examples.
The arithmetic motivic Poincaré series of a variety $V$ defined over a field of characteristic zero, is an invariant of singularities which was introduced by Denef and Loeser by analogy with the Serre-Oesterlé series in arithmetic geometry. They proved that this motivic series has a rational form which specializes to the Serre-Oesterlé series when $V$ is defined over the integers. This invariant, which is known explicitly for a few classes of singularities, remains quite mysterious. In this paper we study this motivic series when $V$ is an affine toric variety. We obtain a formula for the rati…
To study singularities on complex varieties we study Poincaré series of filtrations that are defined by discrete valuations on the local ring at the singularity. In all previous papers on this topic one poses restrictions on the centers of these valuations and often one uses several definitions for Poincaré series. In this article we show that these definitions can differ when the centers of the valuations are not zero-dimensional, i.e. do not have the maximal ideal as center. We give a unifying definition for Poincaré series which also allows filtrations defined by valuations that are all non…
\'Akos K.\ Matszangosz, HUN-REN Alfr\'ed R\'enyi Institute of Mathematics, Re\'altanoda utca 13-15, 1053 Budapest, Hungary；L\'aszl\'o M. Feh\'er, Department of Analysis, Institute of Mathematics, E\"otv\"os Lor\'；University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary；HUN-REN Alfr\'ed R\'enyi Institute of Mathematics, Re\'altanoda utca 13-15, 1053 Budapest, Hungary
We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which expresses these coefficient functions in terms of Jeffrey--Kirwan residues. This is used to prove the uniqueness of the quasipolynomiality inside a special cone, the structure of the counting function in terms of the graph and construction for a polynomial generalization of the Seib…
We classify the singularities of a surface ruled by conics: they are rational double points of type $A_n$ or $D_n$. This is proved by showing that they arise from a precise series of blow-ups of a suitable surface geometrically ruled by conics. We determine also the family of such surfaces which are birational models of a given surface ruled by conics and obtained in a "minimal way" from it.
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 Plücker 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 mode…
A finite subgroup of ${\rm SL}_2(\CC)$ defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity. Here we consider some non-abelian finite subgroups $G$ of ${\rm SL}_3(\CC)$. They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that w…
A quasi-ordinary polynomial is a monic polynomial with coefficients in the power series ring such that its discriminant equals a monomial up to unit. In this paper we study higher derivatives of quasi-ordinary polynomials, also called higher order polars. We find factorizations of these polars. Our research in this paper goes in two directions. We generalize the results of Casas-Alvero and our previous results on higher order polars in the plane to irreducible quasi-ordinary polynomials. We also generalize the factorization of the first polar of a quasi-ordinary polynomial (not necessary irred…
This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface C^2/G with G a finite subgroup of SL(2, C), we conjecture a formula for this generating series in terms of Lie-theoretic data, which is compatible with existing results for type A singularities. We announce a proof of our conjecture for singularities of type D. The generating series in our conjecture can be seen as a specialized character of the basic representation of the corresponding…
In a previous paper, there was defined a multi-index filtration on the ring of functions on a hypersurface singularity corresponding to its Newton diagram generalizing (for a curve singularity) the divisorial one. Its Poincaré series was computed for plane curve singularities non-degenerate with respect to their Newton diagrams. Here we use another technique to compute the Poincaré series for plane curve singularities without the assumption that they are non-degenerate with respect to their Newton diagrams. We show that the Poincaré series only depends on the Newton diagram and not on the defi…
We show that every convergent power series with monomial extended Jacobian ideal is right equivalent to a Thom-Sebastiani polynomial. This solves a problem posed by Hauser and Schicho. On the combinatorial side, we introduce a notion of Jacobian semigroup ideal involving a transversal matroid. For any such ideal we construct a defining Thom-Sebastiani polynomial. On the analytic side, we show that power series with a quasihomogeneous extended Jacobian ideal are strongly Euler homogeneous. Due to a Mather-Yau-type theorem, such power series are determined by their Jacobian ideal up to right equ…
We present a series of examples of pairs of singular semialgebraic surfaces (real semialgebraic sets of dimension two) in ${\mathbb R}^3$ and ${\mathbb R}^4$ which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent, but not ambient Lipschitz equivalent. For each singular semialgebraic surface $S\subset {\mathbb R}^4$, we construct infinitely many semialgebraic surfaces which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent to $S$, but pairwise ambient Lipschitz non-equivalent.
The purpose of this paper is to extend the notions of generalised Poincaré series and divisorial generalised Poincaré series (of motivic nature) introduced by Campillo, Delgado and Gusein-Zade for complex curve singularities to curves defined over perfect fields, as well as to express them in terms of an embedded resolution of curves.
Generalized analytic functions over generalized analytic manifolds are build from sums of convergent real power series with non-negative real exponents (and some well-ordering condition on the support). In a paper by Martín-Villaverde, Rolin and Sanz-Sánchez it is established a result of local reduction of singularities for such a functions. In this paper we deal with a first approach of the global problem. Namely, we prove that a germ of generalized analytic function can be transformed by a finite sequence of blowing-ups with closed centers into a function which is locally of monomial type wi…
This is the first of a series of papers studying real algebraic threefolds using the minimal model program. The main results are outlined in Part II. The present part I. contains the necessary preliminary work concerning terminal singularities. First I give standard forms for 3-dimensional terminal singularities over arbitrary fields. This is then used to develop a fairly complete topological classification over the reals.
In this survey, we review part of the theory of superisolated surface singularities (SIS) and its applications including some new and recent developments. The class of SIS singularities is, in some sense, the simplest class of germs of normal surface singularities. Namely, their tangent cones are reduced curves and the geometry and topology of the SIS singularities can be deduced from them. Thus this class \emph{contains}, in a canonical way, all the complex projective plane curve theory, which gives a series of nice examples and counterexamples. They were introduced by I. Luengo to show the n…
In the classical case of irreducible smooth algebraic curves every genus $2$ curve is hyperelliptic, or in other words there is a complete linear series $g_2^1$ on them. On the other hand if $g > 2$, then a generic smooth curve of genus $2$ is nonhyperelliptic. In this article we investigate the situation of normal surface singularities, so we fix a resolution graph $\mathcal{T}$ and a generic singularity with resolution $\tX$ corresponding to it in the sense of \cite{NNII}. We consider an integer effective cycle $Z$ on the resolution $\tX$ and investigate the existence of a complete linear se…
We compare three naturally occurring multi-indexed filtrations of ideals on the local ring of a Newton nondegenerate hypersurface surface singularity with rational homology sphere, which in many cases are all distinct. These are the divisorial, the order, and the image filtrations. These filtrations are indexed by the lattice associated with a toric partial resolution of the singularity, or equivalently, the free abelian group generated by the compact facets of the Newton polyhedron. We prove that there exists a top dimensional cone contained in the Lipman cone having the property that the thr…
Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring O_{P,X} at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if O_{P,X} is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincare series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the gene…
We construct the analytic lattice cohomology associated with the analytic type of any complex normal surface singularity. It is the categorification of the geometric genus of the germ, whenever the link is a rational homology sphere. It is the analytic analogue of the topological lattice cohomology, associated with the link of the germ whenever it is a rational homology sphere. This topological lattice cohomology is the categorification of the Seiberg--Witten invariant, and conjecturally it is isomorphic with the Heegaard Floer cohomology. We compare the two lattice cohomologies: in some simpl…
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 groups for which the high-dimensional versions of McKay correspondence are valid. In the paper we consider this question in the case of abelian finite subgroups of SL(r,C) by using techniques from toric and discrete geometry. We give two necessary existence conditions, involving the Hilbert basis elements of the cone supporting the junior simplex, and an Up…
