Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with isolated singularities.
A concrete application is for surfaces with geometric genus $p_g=5$: the canonical model is embedded in $\mathbb{P}^4$ if and only if we have a complete intersection of type $(2,4)$ or $(3,3)$.
In this paper, we study the coherence of a higher rank analogue of a multiplier ideal sheaf. Key tools of the study are Hörmander's $L^2$-estimate and a singular version of a Demailly--Skoda type result.
Let $X$ be a Hermitian complex space of pure dimension $n$ with isolated singularities. In the present paper, we give a natural resolution for the canonical sheaf of square-integrable holomorphic $n$-forms with Dirichlet boundary condition on $X$. As application, we obtain an explicit smooth model for the $L^2$-$\bar{\partial}$-cohomology, including natural resolutions for sheaves of $\bar{\partial}$-closed (holomorphic) $L^2$-functions.
Using the semicontinuity of the Milnor number and the Łojasiewicz exponent, we give a simple proof of Yau's characterization of isolated homogenous hypersurface singularities.
Normally one assumes isolated surface singularities to be normal. The purpose of this paper is to show that it can be useful to look at nonnormal singularities. By deforming them interesting normal singularities can be constructed, such as isolated, non Cohen-Macaulay threefold singularities. They arise by a small contraction of a smooth rational curve, whose normal bundle has a sufficiently positive subbundle. We study such singularities from their nonnormal general hyperplane section.
In this paper we review the classification of isolated quotient singularities over the field of complex numbers due to H. Zassenhaus, G. Vincent, and G. A. Wolf. As an application we describe Gorenstein isolated quotient singularities over C, generalizing a result of K. Kurano and S. Nishi.
This note provides a connection between the Hodge spectrum and the Hodge ideals recently developed by Mustaţǎ--Popa in the case of weighted homogeneous isolated singularities.
We give a classification of all Delsarte surfaces with only ADE singularities. Using this we give closed formulas for the Picard numbers of such surfaces.
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.
In this article we apply the results in the article "On Isolated Real Singularities I" to the study of real $ADE$-singularities. We show that said results enables us to find the homology groups of the Milnor fibres of real $ADE$-singularities of curves. As a corollary we show how to deduce the same for $ADE$-singularities in higher codimension, giving an almost complete answer in this case. This is followed by a discussion of the classification problem of simple real singularities.
By the fundamental work of Griffiths one knows that, under suitable assumption, homological and algebraic equivalence do not coincide for a general hypersurface section of a smooth projective variety $Y$. In the present paper we prove the same result in case $Y$ has isolated singularities.
Gorenstein isolated quotient singularities of odd prime dimension are cyclic. In the case where the dimension is bigger than 1 and is not an odd prime number, then there exist Gorenstein isolated non-cyclic quotient singularities.
We prove that the boundaries of the Milnor fibers of smoothings of non-isolated reduced complex surface singularities are graph manifolds. Moreover, we give a method, inspired by previous work of Némethi and Szilard, to compute associated plumbing graphs.
We give a criterion to test geometric properties such as Whitney equisingularity and Thom's $a_f$ condition for new families of (possibly non-isolated) hypersurface singularities that "behave well" with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.
Homological index for 1-forms and a Milnor number for isolated singularities
受X的启发，我们引入了具有孤立奇点的复解析簇芽上的全纯1-形式的同调指数的概念。Gómez-Mont和G.- M.格鲁埃尔对于孤立的完全相交奇点，它与两个作者早先定义的指数一致。从这个指数中减去另一个称为径向的指数，我们得到一个不依赖于1-形式的奇点不变量。对于孤立的完全相交奇点，这个不变量与Milnor数一致。我们计算任意曲线奇点的这个不变量，并将其与R. O. Buchweitz和G. M.格鲁埃尔对于这样的奇点。
We study the topological triviality and the Whitney equisingularity of a family of isolated determinantal singularities. On one hand, we give a Lê-Ramanujam type theorem for this kind of singularities by using the vanishing Euler characteristic. On the other hand, we extend the results of Teissier and Gaffney about the Whitney equisingularity of hypersurfaces and complete intersections, respectively, in terms of the constancy of the polar multiplicities.
In this note we provide a formula for the Tjurina number of a join of isolated hypersurface singularities in separated variables. From this we are able to provide a characterization of isolated hypersurface singularities whose difference between the Milnor and Tjurina numbers is less or equal than two arising as the join of isolated hypersurface singularities in separated variables. Also, we are able to provide new upper bounds for the quotient of Milnor and Tjurina numbers of certain join of isolated hypersurface singularities. Finally, we deduce an upper bound for the quotient of Milnor and …
We prove that if $ϕ:(X,0)\to (X,0)$ is a finite endomorphism of an isolated singularity such that $\operatorname{deg}(ϕ)\geq 2$ and $ϕ$ is étale in codimension 1, then $X$ is $\mathbb{Q}$-Gorenstein and log canonical.
We study the Milnor monodromies of non-isolated hypersurface singularities and show that the reduced cohomology groups of the Milnor fibers are concentrated in the middle degree for some eigenvalues of the monodromies. As an application of this result, we give an explicit formula for some parts of their Jordan normal forms.
We extend the results from the previous paper by A. Frühbis-Krüger and the author [arXiv:1501.01915] to the vanishing topology of those singularities in the title. Studying the case of possibly non-isolated singularities in the Tjurina- transform, we reveal that in dimension 3 and 2 there always is exactly one special vanishing cycle in degree 2 closely related to the determinantal structure of the singularity.
We investigate the existence of complete intersection threefolds $X \subset \mathbb{P}^n$ with only isolated, ordinary multiple points and we provide some sufficient conditions for their factoriality.
Isolated Cohen-Macaulay codimension 2 singularities share many common features with isolated complete intersection singularities, but they also exhibit some striking new behaviour. One such instance was recently observed by Damon and Pike in their study of the vanishing topology and Euler characteristic, where they took this class of singularities as examples. In this article, we explore their findings further by determining the Betti numbers explicitly and explain the new phenomena. An important tool here is the Tjurina modification relating a Cohen-Macaulay codimension 2 singularity to a fin…
In an unpublished lecture note, J. Briançon observed that if $\{f_t\}$ is a family of isolated complex hypersurface singularities such that the Newton boundary of $f_t$ is independent of $t$ and $f_t$ is non-degenerate, then the corresponding family of hypersurfaces $\{f_t^{-1}(0)\}$ is Whitney equisingular (and hence topologically equisingular). A first generalization of this assertion to families with non-isolated singularities was given by the second author under a rather technical condition. In the present paper, we give a new generalization under a simpler condition.
We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity. They are used to study the pull-back of holomorphic 1-forms on an isolated complete intersection curve singularity under the normalization morphism. We wonder whether the Milnor number $μ$ and the Tjurina number $τ$ of any isolated plane curve singularity satisfy the inequality $3μ<4τ$.
In this paper we study the deformation and Q-Gorenstein deformation theory of schemes with non-isolated singularities. We obtain obstruction spaces for the existence of deformations and also for local deformations to exist globally. Finally we obtain explicit criteria in order for a pure and reduced scheme of finite type over a field $k$ to have smoothings and Q-Gorenstein smoothings.
Let X be a complex projective variety of dimension n with only isolated normal singularities. In this paper we prove, using mixed Hodge theory, that if the link of each singular point of X is (n-2)-connected, then X is a formal topological space. This result applies to a large class of examples, such as normal surface singularities, varieties with ordinary multiple points, hypersurfaces with isolated singularities and more generally, complete intersections with isolated singularities. We obtain analogous results for contractions of subvarieties.
We study the essentially isolated determinantal singularities (EIDS), defined by W. Ebeling and S. Gusein-Zade, as a generalization of isolated singularity. We prove in dimension $3$ a minimality theorem for the Milnor number of a generic hyperplane section of an EIDS, generalizing previous results by J. Snoussi in dimension $2$. We define strongly generic hyperplane sections of an EIDS and show that they are still EIDS. Using strongly general hyperplanes, we extend a result of Lê D. T. concerning constancy of the Milnor number.
We give a method to investigate isolated log canonical singularities with index one which are not log terminal. Our method depends on the minimal model program. One of the main purposes is to prove that our invariant coincides with Ishii's Hodge theoretic invariant.
In this survey we overview the different approaches and solutions of a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers.
We study the Euler obstruction of essentially isolated determinantal singularities (EIDS). The EIDS were defined by W. Ebeling and S. Gusein-Zade, as a generalization of isolated singularity. We obtain some formulas to calculate the Euler obstruction for the determinantal varieties with singular set an ICIS.
We prove that there are just two types of isolated singularities of special Kähler metrics in real dimension two provided the associated holomorphic cubic form does not have essential singularities. We also construct examples of such metrics.
We show that isolated surface singularities which are non-normal may have Milnor fibers which are non-diffeomorphic to those of their normalizations. Therefore, non-normal isolated singularities enrich the collection of Stein fillings of links of normal isolated singularities. We conclude with a list of open questions related to this theme.
We show the existence of a full exceptional collection in the graded stable derived category of a Gorenstein isolated quotient singularity using a result of Orlov (arXiv:math/0503632). We also show that the equivariant graded stable derived category of a Gorenstein Veronese subring of a polynomial ring with respect to an action of a finite group has a full strong exceptional collection, even if the corresponding quotient singularity is neither isolated nor Gorenstein.
We prove that the class of log canonical rational singularities is closed under the basic operations of the minimal model program. We also give some supplementary results on the minimal model program for log canonical surfaces.
Let a finite group G act linearly on a finite dimensional vector space V over an algebraically closed field k of characteristic p>2. Assume that the quotient V/G is an isolated singularity. In the case when p does not divide the order of G, isolated singularities V/G are completely classified and their classification reduces to Zassenhaus-Vincent-Wolf classification of isolated quotient singularities over the field of complex numbers. In the present paper we show that if dimension of V is 3, then also in the modular case (p divides the order of G) classification of isolated quotient singularit…
