M. Kawashima
In this paper, we consider curves of degree 10 of torus type (2,5), C : f_5(x, y)^2 + f_2(x, y)^5 = 0. Assume that f_2(0, 0) = f_5(0, 0) = 0. Then O = (0, 0) is a singular point of C which is called an inner singularity. In this paper, we give a topological classification of singularities of (C,O).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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 homogeneous generators of the Cox ring in a more general setup, based on the multigraded Castelnuovo-Mumford regularity and Kawamata-Viehweg vanishing.
We classify the canonical threefold singularities that allow an effective two-torus action. This extends classification results of Mori on terminal threefold singularities and of Ishida and Iwashita on toric canonical threefold singularities. Our classification relies on lattice point emptiness of certain polytopal complexes with rational vertices. Scaling the polytopes by the least common multiple $k$ of the respective denominators, we investigate $k$-emptiness of polytopes with integer vertices. We show that two dimensional $k$-empty polytopes either are sporadic or come in series given by F…
Using Altmann-Hausen-Suss description of $\mathbb{T}$-varieties via divisorial fans and Kóvacs-Schwede-Smith characterization of Du Bois singularities, we study Cohen-Macaulay Du Bois $\mathbb{T}$-singularities of complexity one. We exhibit cohomological criteria for a $\mathbb{T}$-variety to be Cohen-Macaulay and Du Bois in terms of polyhedral divisors. We give an example of a Cohen-Macaulay Du Bois singularity of complexity one which does not have rational singularities.
We propose a method to compute a desingularization of a normal affine variety X endowed with a torus action in terms of a combinatorial description of such a variety due to Altmann and Hausen. This desingularization allows us to study the structure of the singularities of X. In particular, we give criteria for X to have only rational, (QQ-)factorial, or (QQ-)Gorenstein singularities. We also give partial criteria for X to be Cohen-Macaulay or log-terminal. Finally, we provide a method to construct factorial affine varieties with a torus action. This leads to a full classification of such varie…
This is an announcement of the following construction: given an integral affine manifold $B$ with singularities, we build a topological space $X$ which is a torus fibration over $B$. The main new feature of the fibration $X\to B$ is that it has the discriminant in codimension 2.
The Classification of Rigid Torus Quotients with Canonical Singularities in Dimension Three
本文在双全纯与微分同胚意义下对具有孤立典范奇点的刚性三维环面商空间进行了精细分类，并补充了具有 Gorenstein 奇点的 III₀ 型 Calabi-Yau 三维流形分类。
The second author classified configurations of the singularities on tame sextics of torus type. In this paper, we give a complete classification of the singularities on irreducible sextics of torus type, without assuming the tameness of the sextics. We show that there exists 121 configurations and there are 5 pairs and a triple of configurations for which the corresponding moduli spaces coincide, ignoring the respective torus decomposition.
The main purpose of this paper is to prove the existence of the moduli space parameterizing the embedded curve singularities of $(k^N,0)$ with an admissible Hilbert polynomial and to study its basic properties.
We present an algorithm for computing instanton numbers of curve singularities. A comparison is made between these and some other invariants of curve singularities. The algorithm has been implemented in the symbolic computer algebra program Macaulay2, and can be downloaded from http://www.math.nmsu.edu/\~{}iswanson/instanton.m2.}
We give a complete classsification of reduced sextics of torus type with configurations of the singularities and the geometry of the components.
A criterion is given in order that the ideals of a one branch curve singularity form at most 2-parameter families. Namely, we present a list of plane curve singularities from the Arnold's classification which are the smallest among all one branch singularities having at most 2-parameter families of ideals.
We study the pull-back of regular 1-forms on a complex irreducible plane curve singularity under the normalization morphism.
We study reflexive modules over one dimensional Cohen-Macaulay rings. Our key technique exploits the concept of $I$-Ulrich modules.
Fixed a point O on a non-singular surface S and a complete mO-primary ideal I in its local ring, the curves on the surface X obtained by blowing-up I are studied in terms of the base points of I. Criteria for the principality of these curves are obtained. New formulas for their multiplicity, intersection numbers and order of singularity at the singularities of X are given. The semigroup of branches going through a sandwiched singularity is effectively determined, too.
Plane curve singularities and Fitting ideals
研究非拟齐次平面曲线奇点的Tjurina理想相关的Fitting理想，发现Milnor数与Tjurina数之差不超过2时的特殊性质。
We consider the parameterization ${\mathbf{f}}=(f_0,f_1,f_2)$ of a plane rational curve $C$ of degree $n$, and we want to study the singularities of $C$ via such parameterization. We do this by using the projection from the rational normal curve $C_n\subset \mathbb{P}^n$ to $C$ and its interplay with the secant varieties to $C_n$. In particular, we define via ${\mathbf{f}}$ certain 0-dimensional schemes $X_k\subset \mathbb{P}^k$, $2\leq k\leq (n-1)$, which encode all information on the singularities of multiplicity $\geq k$ of $C$ (e.g. using $X_2$ we can give a criterion to determine whether …
In this paper, we compute the fundamental group of the complement of linear torus curves of maximal contact and we show that it is isomorphic that of generic linear torus curves.
We consider families of parameterizations of reduced curve singularities over a Noetherian base scheme and prove that the delta invariant is semicontinuous. In our setting, each curve singularity in the family is the image of a parameterization and not the fiber of a morphism. The problem came up in connection with the right-left classification of parameterizations of curve singularities defined over a field of positive characteristic. We prove a bound for right-left determinacy of a parameterization in terms of delta and the semicontinuity theorem provides a simultaneous bound for the determi…
Limit distribution of Hodge spectral exponents of irreducible plane curve singularities
本文研究不可约平面曲线奇点的Hodge谱指数分布，通过数值不变量给出其与连续分布差值的闭式公式，并刻画极限分布为连续分布的曲线族及占优值区间。
In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on singularities with a torus action of complexity 1, threefold singularities, and hypersurface singularities. Given a real value v>0, we prove that the class of K-semistable threefold singularities with normalized volume at least v forms a bounded family. Analogous statements are proved in the case of n-dimensional complexity-1 and n-dimensional hypersurface singularities for arbitary n. In the general case of klt singularities, i.e. without the assumption on K-semistability, we show th…
Functorial resolution by torus actions
提出利用环面作用在带SNC除子的光滑簇上，通过新引入的沿光滑加权中心的配边爆破，实现特征零下簇的快速嵌入解消与理想主化，并应用于正和混合特征奇点的解消。
Using the language of T-varieties, we study torus invariant curves on a complete normal variety $X$ with an effective codimension-one torus action. In the same way that the $T$-invariant Weil divisors on $X$ are sums of "vertical" divisors and "horizontal" divisors, so too is each $T$-invariant curve a sum of "vertical" curves and "horizontal" curves. We give combinatorial formulas that calculate the intersection between $T$-invariant divisors and $T$-invariant curves, and generalize the celebrated toric cone theorem to the case of complete complexity-one $T$-varieties.
In this paper, we compute Alexander polynomials of a torus curve C of type (2, 5), C : f(x, y) = f_2(x, y)^5 + f_5(x, y)^2 = 0, under the assumption that the origin O is the unique inner singularity and f2 = 0 is an irreducible conic. We show that the Alexander polynomial remains the same with that of a generic torus curve as long as C is irreducible.
We prove that if two plane curve singularities are equisingular, then they are topologically equivalent. The method we will use is P.~Fortuny~Ayuso's who proved this result for irreducible plane curve singularities.
We study the classification of plane curve singularities in arbitrary characteristic. We first give a bound for the determinacy of a plane curve singularity with respect to pararametrization equivalence in terms of its conductor. Then we classify parametrization simple plane curve singularities which are irreducible by giving a concrete list of normal forms of equations and parametrizations. In characteristic zero, the classification of parametrization simple irreducible plane curve singularities was achieved by Bruce and Gaffney.
We show the existence of sextics of non-torus type which is a Zariski partner of the tame sextics of torus type with simple singularities.
We study topological zeta functions of complex plane curve singularities using toric modifications and further developments. As applications of the research method, we prove that the topological zeta function is a topological invariant for complex plane curve singularities, we give a short and new proof of the monodromy conjecture for plane curves.
We study families of $K3$ surfaces obtained by double covering of the projective plane branching along curves of $(2,3)$-torus type. In the first part, we study the Picard lattices of the families, and a lattice duality of them. In the second part, we describe a deformation of singularities of Gorenstein $K3$ surfaces in these families.
We generalize a theorem of Delzant classifying compact connected symplectic manifolds with completely integrable torus actions to certain singular symplectic spaces. The assumption on singularities is that if they are not finite quotient then they are isolated.
We classify simple parametrisations of complex curve singularities. Simple means that all neighbouring singularities fall in finitely many equivalence classes. We take the neighbouring singularities to be the ones occurring in the versal deformation of the parametrisation. This leads to a smaller list than that obtained by looking at the neighbours in a fixed space of multi-germs. Our simple parametrisations are the same as the fully simple singularities of Zhitomirskii, who classified real plane and space curve singularities. The list of simple parametrisations of plane curves is the A-D-E li…
Gorenstein curve singularities of genus three
本文分类了亏格三的孤立Gorenstein曲线奇点的解析芽，并建立了它们与阿贝尔微分分层空间连通分支之间的对应关系。
We classify stably simple reducible curve singularities in complex spaces of any dimension. This extends the same classification of of irreducible curve singularities obtained by V.I.Arnold. The proof is essentially based on the method of complete transversals by J.W.Bruce et al.
We obtain some recombination formulae for the spectra of (complex, reduced) plane curve singularities. As an application we prove: a generalization of Durfee's bound; a generalization of Givental's bound; the multiplicity of the curve singularity is determined by its spectrum; for many curve singularities all the multiplicities of exceptional divisors of the resolution are determined by the spectrum; etc.
We present an intersection-theoretical approach to the invariants of plane curve singularities $μ$, $δ$, $r$ related by the Milnor formula $2δ=μ+r-1$. Using Newton transformations we give formulae for $μ$, $δ$, $r$ which imply planar versions of well-known theorems on nondegenerate singularities.
We proved that every rational curves in the primitive class of a general K3 surface of any genus is nodal.
We compute Bernstein-Sato polynomials of some pairs of topologically equivalent plane curve singularities. Some pairs have the same Tjurina number but distinct Bernstein-Sato polynomials, which implies that they are not analytically equivalent.
Pfister and Steenbrink studied punctual Hilbert schemes for irreducible curve singularities. In particular, they investigated the structure of special punctual Hilbert schemes for certain monomial curve singularities. In this paper, we study the punctual Hilbert schemes of all degrees for the curve singularities of types $E_6$ and $E_8$. For our analysis, we introduce computational algorithms to decompose a punctual Hilbert schemes into affine cells. We also use the theory of Gröbner basis and known results about the compactified Jacobian of singular curves to prove our main theorems.
The complete list of reducible sextics of torus type with simple singularities is known in our previous paper. In this paper, we give a complete list of existence and non-existence of Zariski partner sextics of non-torus type corresponding to the list of sextics of torus type.
In this article, we study Cohen-Macaulay modules over non-reduced curve singularities. We prove that the rings $k[[x,y,z]]/(xy, y^q -z^2)$ have tame Cohen-Macaulay representation type. For the singularity $k[[x,y,z]]/(xy, z^2)$ we give an explicit description of all indecomposable Cohen--Macaulay modules and apply the obtained classification to construct explicit families of indecomposable matrix factorizations of $(xy)^2 \in k[[x,y]]$.
We show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularities. In Theorem 1.1 we list integers can be attained as Milnor numbers of a given semi-quasi-homogeneous singularity.
We deform monomial space curves in order to construct examples of set-theoretical complete intersection space curve singularities. As a by-product we describe an inverse to Herzog's construction of minimal generators of non-complete intersection numerical semigroups with three generators.
This paper is the first part of a two part paper which introduces the study of the Whitney Equisingularity of families of Symmetric determinantal singularities. This study reveals how to use the multiplicity of polar curves associated to a generic deformation of a singularity to control the Whitney equisingularity type of these curves.
We give a $δ$-constant criterion for equinormalizability of deformations of isolated (not necessarily reduced) curve singularities over smooth base spaces of dimension $\geq 1$. For one-parametric families of isolated curve singularities, we show that their topologically triviality is equivalent to the admission of weak simultaneous resolutions.
We study adjacency of equisingularity types of planar curve singularities in terms of their Enriques diagrams. For linear adjacency a complete answer is obtained, whereas for arbitrary (analytic) adjacency a necessary condition and a sufficient condition are proved. We also show an example of singular curve of type D' that can be deformed to a curve of type D without D' being adjacent to D.
This article, based on the talk given by one of the authors at the Pierrettefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of plane curve singularities.
We generalize Macdonald's formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to the cohomology of the compactified Jacobian of the curve. The new formula is a consequence of a stronger identity between certain perverse sheaves defined by a family of curves satisfying mild conditions. The proof makes an essential use of Ngo's support theorem for compactified Jacobians and generalizes this theorem to the relative Hilbert scheme of such families. As a consequence, we generalize …
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.
If the first Betti number of the Milnor fibre of a plane curve singularity is zero, then the defining function is equivalent to $x^r$.
In this note, we present a new method for computing fundamental groups of curve complements using a variation of the Zariski-Van Kampen method on general ruled surfaces. As an application we give an alternative (computation-free) proof for the fundamental group of generic $(p,q)$-torus curves.
In a recent paper Chiodo and Farkas described the singular locus and the locus of non-canonical singularities of the moduli space of level curves. In this work we generalize their results to the moduli space $\overline{\mathcal R}_{g,G}$ of curves with a $G$-cover for any finite group $G$. We show that non-canonical singularities are of two types: $T$-curves, that is singularities lifted from the moduli space $\overline{\mathcal M}_g$ of stable curves, and $J$-curves, that is new singularities entirely characterized by the dual graph of the cover. Finally, we prove that in the case $G=S_3$, th…
We characterize plane curve germes non-degenerate in Kouchnirenko's sense in terms of characteristics and intersection multiplicities of branches.
In this paper we give some criteria for a family of generically reduced plane curve singularities to be equinormalizable. The first criterion is based on the $δ$-invariant of a (non-reduced) curve singularity which is introduced by Brücker-Greuel (\cite{BG}). The second criterion is based on the I-equisingularity of a $k$-parametric family ($k\geq 1$) of generically reduced plane curve singularities, which is introduced by Nobile (\cite{No}) for one-parametric families. The equivalence of some kinds of equisingularities of a family of generically reduced plane curve singularities is also studi…
We introduce moduli spaces of quasi-admissible hyperelliptic covers with at worst A and D singularities. The stability conditions for these moduli spaces depend on two parameters describing allowable singularities. By varying these parameters, we interpolate between the stacks of stable limits of A and D singularities and the quotients of the versal deformation spaces of A and D singularities by a natural $\mathbb{G}_m$ action.
We study deformations of plane curve singularities from an analytic point of view and obtain some new concrete results. We show some rather unexpected properties of Puiseux coefficients treated as functions on a suitably defined parameter space. The methods used in paper are very elementary.
Tools for analyzing the intersection curve between a torus and a quadric through projection and lifting
本文提出基于投影与提升的工具，利用结式与子结式多项式理论分析圆环面与不可约二次曲面交线的投影、奇点及双重切线。
In this paper we consider the singularities of the varieties parameterizing stable vector bundles of fixed rank and degree with sections on a smooth curve of genus at least two. In particular, we extend results of Y. Laszlo, and of the second author, regarding the singularities of generalized theta divisors.
Lattice cohomology and the embedded topological type of plane curve singularities
本文提出解析格上同调这一约化曲线奇点新不变量，证明其对不可约平面曲线奇点是完备嵌入拓扑不变量，并在多分支情形下与整Seifert形式比较。
We provide a tool how one can view a polynomial on the affine plane of bidegree $(a,b)$ - by which we mean that its Newton polygon lies in the triangle spanned by $(a,0)$, $(0,b)$ and the origin - as a curve in a Hirzebruch surface having nice geometric properties. As an application, we study maximal $A_k$-singularities of curves of bidegree $(3,b)$ and find the answer for $b\leq 12$.
This paper deals with a complete invariant $R_X$ for cyclic quotient surface singularities. This invariant appears in the Riemann Roch and Numerical Adjunction Formulas for normal surface singularities. Our goal is to give an explicit formula for $R_X$ based on the numerical information of $X$, that is, $d$ and $q$ as in $X=X(d;1,q)$. In the process, the space of curvettes and generic curves is explicitly described. We also define and describe other invariants of curves in $X$ such as the LR-logarithmic eigenmodules, $δ$-invariants, and their Milnor and Newton numbers.
It is a very old and interesting open problem to characterize those collections of embedded topological types of local plane curve singularities which may appear as singularities of a projective plane curve C of degree d. The goal of the present article is to give a complete (topological) classification of those cases when C is rational and it has a unique singularity which is locally irreducible (i.e. C is unicuspidal) with one Puiseux pair.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。