Questions tagged [singularity-theory]

This tag is for questions relating to Singularity Theory. In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes.

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29 views

Is the desingularization of this surfaces of general type?

I have a family of deformation $\mathcal{X}\to B$ of a surface $S$. The surface has a finite number of singular point and it is normal. There is a divisor $D\subset B$ such that the surface $\mathcal{...
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1answer
25 views

Prove that for for an arbitrary linear operator all eigenvalues lie in a circular ring.

Prove that for for an arbitrary linear operator all eigenvalues lie in a circular ring. {z $\in$ $С$ | $\sigma_n(A)$ $\leq$ $|z|$ $\leq \sigma_1(A)$}
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30 views

Topology of singular fibers only determined by the singular types?

I am interested in the topology of singular fibers. I came across the following two questions (in the complex geometry setting): Let $X \to T$ be a proper submersion except for finitely many ...
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1answer
68 views

Unknot: embedded vs immersed bounding disk

Suppose that we have a knot $K\subset \mathbb{S}^3$. If $K$ bounds* an embedded 2-disk then $K$ is the unknot. But what happens if $K$ bounds an immersed 2-disk? The immersed disk generically will ...
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2answers
96 views

the range of the entire function $ e^z(1+\cos\sqrt{z} ) $ Picard's Great theorem for

Let $$f(z) = e^z (1+\cos\sqrt{z} ) $$ $\Omega=\{z\in\Bbb C: |z|\gt r\}$, $r\gt 0$. What is $f(\Omega)$? where $\sqrt{z}=\exp{(\text{Log }z/2)}, \text{Log }z=\log|z|+i\arg z,\arg z\in(-\pi,\...
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1answer
111 views

Tangent cone of a non-isolated singular point

Assume $X\subset \mathbb P^n$ is a variety (Edit: let's say $X$ is a hypersurface in $\mathbb P^n$, as pointed out in the comment) and $x\in X$ is a singular point which is not isolated. Intuitively, ...
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1answer
49 views

Isolated critical points and the Milnor number

I was looking at the Wikipedia page of the Milnor number found here, and specifically tried to work through proving for myself one of its stated properties. It mentions the Milnor number is finite if ...
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1answer
70 views

Exceptional divisor of the blow-up of affine cone at the vertex

Let $f(x) \in \mathbb{C}[X_1,...,X_n]$ be a homogeneous polynomial in $n$ variables such that the zero locus $V$ of $f$ in $\mathbb{C}^n$ is singular only at the origin. Denote by $\pi:\widetilde{V} \...
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1answer
38 views

Reference request: Knots that don't come from Milnor spheres.

In Milnor's book "Hypersurface singularities" He discusses shortly knots that arrive as Milnor spheres of algebraic curves, i.e knots that are the intersection of a $3$ sphere around a singular point ...
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27 views

Show that if $f \sim^{\mathcal{R}} g$, then $\text{ord } (f) = \text{ord } (g)$.

Problem: Let $f,g \in \mathbb{C}[[\mathrm{x}]] = \mathbb{C}[[x_1,\dots,x_n]]$. Show that if $f \sim^{\mathcal{R}} g$ then $\text{ord } (f) = \text{ord } (g)$. My attempt: Follow the definition, we ...
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32 views

Is this problem another state of the Weierstraß Division Theorem (WDT)?

Problem: Let $\mathfrak{m} = \langle x_1,x_2,\dots,x_n \rangle \subset K[[x_1,x_2,\dots,x_n]]$, let $f \in K[[x_1, \dots, x_n]]$ be $x_n\mkern-1.5mu$-general of order $b$, $\text{ord}(f)=b$ and $g \...
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25 views

Complex structure deformations of toric Calabi-Yau 3-folds

If $X$ is a toric Calabi-Yau 3-fold (so, it is in particular, non-compact), is it necessarily true that $h^{2,1}(X)$ vanishes? (Such a CY 3-fold is termed ``rigid''.) For example, the resolved ...
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175 views

Can we approximate a vector field on the plane with non-vanishing vector fields in $L^2$?

Let $V$ be a compactly-supported smooth vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated. Does there exist a sequence ...
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14 views

General formalism for singularities?

In mathematics, we have the notion of singularity. There are known (and well-studied) classes, but is it correct to say that there is no general definition of singularity? "General" in the sense that ...
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1answer
36 views

What is the name of this type of object?

I am interested in the following type of objects (I will describe them later), but because I am very new to the subject of differential geometry, and I just have a very basic notion of the concepts, I ...
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46 views

Calculating infinitely near points of a curve singularity

Let $f(x,y)$ be a polynomial in $\mathbb{Q}[x,y]$ with a singular point at the origin $(0,0)$. Currently I am writing a Maple program to compute its iterated blow-up embedded resolution as a hierarchy ...
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2answers
85 views

Deciding whether a point on a surface is singular

Consider $S^2 \in \mathbb{R}^2$. It is well known that it is a regular surface. We can parametrize (a patch of it) by (EDIT: this seems wrong, see comment) $$x(u,v) = \cos u \cos v$$ $$y(u,v) = \sin ...
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35 views

How many extra generators in the normalization of $\mathbb{C}[x,y]/f(x,y)$ vs. how many blow ups are needed.

Let $X=Z(f)$ be a curve in the affine plane with a unique singular point $(0,0)$. Let's keep in mind the examples $f=y^2-x^5,\ f=y^2-x^2-x^3 $. Now we want to calculate the normalization of $R=k[x,y]/...
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53 views

Topology of the link of a rational hypersurface singularity

Let $V\subseteq \mathbb{C}^{n+1}$ be an affine algebraic hypersurface, i.e. a zero set of a polynomial in $n+1$ complex variables. Let $S$ be a small sphere around the origin $0\in \mathbb{C}^{n+1}$ ...
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1answer
68 views

Milnor Number for holomorphic map germ

Definition: Let $f : (\mathbb{C}^{n}, p) \longrightarrow (\mathbb{C}^{n}, q)$ be a holomorphic map germ. The multiplicity of $f$ at $p$, or Milnor number de $f$ at $p$, noted $\mu_{p}(f)$, is the ...
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39 views

vanishing of higher de Rham complex/cohomology of scheme without regularity/smoothness condition

If $f:X\to Y$ is a morphism of schemes of relative dimension $d$, and $\Omega_{X/Y}$ the sheaf of relative Kahler differentials, then we have the associated de Rham complex $$\mathcal{O}_X\to \Omega^...
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1answer
119 views

Condition for polynomials to be proper

Let $\Bbbk\in \left\{ \mathbb R,\mathbb C \right\}$. Suppose $\mathbb \Bbbk^n\overset{f}{\to} \Bbbk$ is a homogeneous polynomial map satisfying the following condition: the fiber of $f$ containing the ...
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35 views

A generic tangent plane gives non-degenerate tangent cone and etc.

Let $X\subset \mathbb P^n=\mathbb P^n_\mathbb C$ be a smooth hypersurface of degree $d$, $d>1$. For any $x\in X$, the tangent plane $H$ intersects with $X$ at $x$, thus $X\cap H$ is singular at $x$....
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1answer
164 views

Milnor Number of real and imaginary parts of holomorphic germs?

By performing some computations using the Singular software, I've noticed the following pattern: if $\mu$ is the Milnor Number of a holomorphic germ $f\in \mathcal{O}_n$ at the origin, then the Milnor ...
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76 views

algebraic de Rham cohomology of normal crossing singularities

For a variety $X/k$ the standard way of defining de Rham cohomology $H^i_{\mathrm{dR}}(X)$ is as the hypercohomology $\mathbb{H}^i(\Omega^{\bullet}_{X/k})$ of the de Rham complex, and this requires $X$...
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1answer
63 views

Same homotopy type as circle

Let $S=\{(z_1,z_2,\ldots, z_n)\in \mathbb{C}^n: |z_1|^2+|z_2|^2+\ldots+|z_n|^2=1\}$ let $K=\{(z_1,z_2,\ldots, z_n)\in S: |z_1|^2+|z_2|^2+\ldots+|z_{n-1}|^2=1\}$. Prove that $S\setminus K=M$ has same ...
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45 views

Extension of holomorphic function to exceptional divisors

Let $(X,0) \subset \mathbb{C}^n$ be complex analytic set with an isolated singularity. Let $f:(X,0) \to (\mathbb{C},0)$ be a germ of a reduced holomorphic function germ defined on it. Let $\pi:E \to \...
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1answer
361 views

asymptotic expansion for integral when integrand is function of limits

I have the following integral: $$f(x) = \int_{1}^{x} \dfrac{t^{-\alpha}}{(x - t)^{\frac{3}{2}} } \exp{\left( \dfrac{-A^2 {t}^2}{(x - t)} \right)} dt, A > 0, \alpha > 0$$ I want to calculate the ...
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1answer
118 views

Definition of $\operatorname{Pic}^0(V)$ for $V$ a singular variety

How does one define the $\operatorname{Pic}^0(V)$ for $V$ being a singular, not necessarily normal variety? Until now the approach I found by searching Google is to prove that the Picard functor is ...
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1answer
48 views

Sparse matrix computational difficulties

I am a computer science student. But I start writing here because my problem is not code related. It is purely computational related. I am trying to calculated inverse of a large (e.g., $2000 \times ...
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34 views

Injective Images of Arcs and Trees Under Meromorphic Maps

This question is pretty, and easy to state. Given a non-constant meromorphic function $f$ on the unit disc (or any complex surface) with only finitely many isolated singularities $x_1, \dots, x_n$, ...
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2answers
68 views

Local equation(s) for transversal (surface) singularities

I would like to write down local equation(s) for a surface $S\subset \mathbb C^4$ whose singular locus is itself a singular curve $C$, say $C=\{z=w=x^2-y^2=0\}$, where $(x,y,z,w)$ are the coordinates ...
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1answer
254 views

Small contractions as blow ups

I am trying to learn a bit about birational morphisms: $f: X\rightarrow Y$, between (projective) normal varieties. In particular, it is well known that every such morphism is a blow-up (e.g ...
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1answer
22 views

complex function -singular points of function

I'm trying to find which kind of singularity infinity is of the function $$ f(z) = \frac{\cos(z + i) - 1}{(z + i)^4} $$ thanks
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23 views

characterization of singularities of dynamic system

Is there any characterization of singularities of a dynamical system in ${R}^2$? How to characterize singularities of vector fields? Can You suggest any book which explains geometrically? Thanks, ...
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1answer
190 views

Find and classify singularities, calculate their residues

$f(z) = \frac{(cos(z)-1)sin(z)}{e^{3z}z^4(z-\pi)^2}$ Approach Singularities of denominator at $z=0$ of order $4$, $z=\pi$ of order $2$. Singularities of numerator at $z=0$ of order $3$. Gives overall ...
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1answer
78 views

Are isolated singularities always cone-like?

Assume $(X,x_0)$ is a pointed topological space such that $X \backslash\{x_0\}$ is a manifold. Does there always exist a neighbourhood $U$ of $x_0$ which is homeomorphic to the cone $cN := (0,1]\times ...
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48 views

Intersection form of a Pseudomanifold

Do pseudomanifolds have an intersection form? They have a volume cycle so it seems like you can use this to define a pairing on $H^{middle}(X,\mathbb{Z})$, however I've only see references to the ...
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33 views

$(p,q)$-forms for analytical spaces?

Let $X = X_{\mathbb C}$ be a complex analytical variety. There is a natural functor from complex analytical varities to real analytical varieties $$ (X, \mathcal{O}_X) \mapsto (X_{\mathbb{R}}, \...
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56 views

Describe the singularity of $\frac{xy}{x^2 + y^2}$ near $(x,y) = (0,0)$

in calculus class we are shown the function $ f(x,y) = \frac{xy}{x^2 + y^2} $ is not $C^\infty$ at $(x,y) = (0,0)$. However if we exclude the origin, we can define a surface: $$ \left\{ \left( x,y,...
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34 views

Upper bound for the dimension of a singularity locus

I am investigating the singularities of a manifold defined as follows, $$ M =\{x \in \mathbb{R}^n \, |\, g_1(x)=u_1, \cdots, g_p(x)=u_p\} $$ where $g_i:\mathbb{R}^n \rightarrow \mathbb{R}$, $i=1,\...
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110 views

Whitney conditions on a stratified space vs smooth manifold

I have read that the Whitney conditions (WC) are sufficient conditions to ensure the continuity of the tangent space on a stratified space. Thus a stratified space with WC looks as a smooth manifold. ...
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328 views

holomorphic vector fields tangent to a hypersurface

Let $(V,0)\subset (\mathbb{C}^n,0)$, $n\geq 3$ a (germ of) hypersurface given by $\{f=0\}$, $f$ holomorphic. A (germ of) holomorphic vector field $X$ leaves $V$ invariant if ${\rm d}f(X)$ belongs to ...
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1answer
259 views

Multiplicity of a variety at a point

Let $X\subset\mathbb{P}^n$ be a variety with associated homogeneous ideal $I(X) = (f_1,...,f_r)$ where $f_i = f_i(x_0,...,x_n)$ is an homogeneous polynomial, and let $p\in\mathbb{P}^n$ be a point. ...
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46 views

Determining wall crossings in toric geometry

In toric geometry, we can describe manifolds by specifying a fan, which is roughly a collection of cones (of various dimensions) in an $n$-dimensional lattice. Many topological properties can be ...
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92 views

The corresponding relation between singularities and continued fraction

As we know, there are classes of continued fractions, finite CF / infinite CF, periodic CF/ nonperiodic CF, bounded/ unbounded. Or do classification according to the number it corresponds. And we ...
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36 views

Singularities of power series with coefficients over $\mathbb{N}$

Given power series with coefficients over $\mathbb{N}$, how to find it's singularity and the order of singularities, if it is with natural boundary, how to find it's order of singularities, and ...
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42 views

Singularities from topological viewpoint

I intuitively understand that \begin{align} f:(-\infty.0)\cup(0,+\infty) &\to \mathbb{R} \\ x &\mapsto \dfrac{1}{x} \end{align} is "singular" at $x=0$, however, is there a topological ...
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1answer
73 views

Is this a removable discontinuity? [closed]

Consider the function $$ f(x):=x\text{ mod }2\pi,~~x\in [0,2\pi). $$ Then obviously, $f$ is not defined for $x=2\pi$ but $\lim_{x^-\to 2\pi}f(x)=0$. I am not sure whether $x=2\pi$ is a removable ...
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191 views

application of Riemann's theorem on removable singularities

This is an exercise from Conway. I managed to prove the first part of (a), but I can't understand the second part of (a). How can it be possible that f(a) belongs to the boundary of Ω? By the Open ...

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