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Questions tagged [singularity-theory]

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1answer
36 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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27 views

How to see link of a divide given in A'campo's paper is link

In the paper "Generic immersions of curves,knots,monodromy and gordian number" by A'Campo definition of a divide and divide of link given as below First of all as P is immersion it is not have to be ...
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0answers
67 views

When does the link of an algebraic singularity determines it algebraic type?

Let $X \subset \Bbb C^n$ be an algebraic hypersurface with an isolated singularity $x$ which is locally irreducible, i.e the local ring $\mathcal O_{X,x}$ is an integral domain (this is a necessary ...
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29 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 \...
3
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1answer
112 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 ...
1
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1answer
68 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 ...
2
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1answer
43 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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0answers
33 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
48 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 ...
3
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1answer
114 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
21 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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22 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
36 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 ...
2
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1answer
56 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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0answers
31 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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0answers
30 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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0answers
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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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0answers
30 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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0answers
73 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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270 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 ...
1
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1answer
155 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. ...
2
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0answers
29 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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0answers
62 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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0answers
35 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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0answers
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
69 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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0answers
141 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 ...
2
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1answer
216 views

Why do we blow-up only closed points on curves

I was reading Cutkosky's Resolution of Singularities and at chapter 3 we endeavor into the resolution of embedded curves through blowing-up. The situation is simple: we have a regular surface $S$ and ...
2
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0answers
92 views

Blowing up lines in projective space

In $\mathbb{A}^3$ we can transform the surface $x^2=y^2z$ to a smooth surface by blowing up the z-axis. What if we want to resolve this surface everywhere in $\mathbb{P}^3$? If we take projective ...
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87 views

What regularization of singularity means?

I have this function, $$F(r) = \frac{1}{r^{6}-1}$$ What does regularization of the singularity mean? I think we can add a small imaginary part to remove the singularity like given below. $$F(r) = \...
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1answer
53 views

What type of singularity is $z=0$

Find the residue of $\frac {\cot(z)\coth(z)}{z^3}$ at $z=0$. In my notebook, i expanded the functions and took the coefficient of $1/z$ as residue, which is $-7/45$ I couldn't recall what type of ...
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0answers
80 views

Why doesn't this point of the quintic have a singular Hessian?

I am considering the quintic: $$ x_1^5 +x_2^5 +x_3^5 +x_4^5 +x_5^5 - 5 x_1x_2x_3x_4x_5=0$$ and it is said (Candelas et al.) that only one singular point which we may as well take to be the ...
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0answers
43 views

Compute Singularity from the given weighted dual graph

For a rational surface singularity, Artin's result guarantees the existence of the fundamental cycle Z. Thinking in the reverse direction, I have the following question(s): Que-1: Given a weighted ...
1
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1answer
123 views

Inverse Laplace transform of a function with removable singularities

While trying to figure out the inverse Laplace transform of $f(z)=\frac{exp(-z)(z+2) +z-2}{z^{3}}$ by using the sum of residues of the function $f(z)exp(zt)$ (from the inversion theorem) I have ...
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0answers
45 views

Resolving singularities in elliptically fibered fourfolds

Consider a Calabi-Yau fourfold given as an elliptic fibration $\pi : X_4 \to B_3$ over a complex threefold. The discriminant locus $\{\Delta=0\}$ describes the locus in $B_3$ over which the elliptic ...
1
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1answer
194 views

Application of residue theorem for improper integrals

While i am reading one example in the book, i came across the book teaching me how to evaluate $\int_{-\infty}^{\infty}\dfrac{\sin x}{x}dx$ by using residue theorem. However, while they say we still ...
1
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2answers
89 views

Singularities of $\frac{z^2}{(z^2+1)^2(z^2+2z+2)}$.

Let $f(z) = \frac{z^2}{(z^2+1)^2(z^2+2z+2)}$. Find singularities of this function. For each singularity determine if it is removable, a pole (if a pole determine it's order,) or essential. I ...
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0answers
106 views

Classification of Isolated singularities

I was wondering if there is a fast and easy way to classify isolated singularities rather than investigating the behaviour of the limit or the Laurent series. I understand that such shortcuts might ...
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2answers
251 views

If $z=a$ is not the removable singularity of $f$, show that $e^{f(z)}$ has essential singularity at $z=a$.

Let $f$ be analytic on $0<|z-a|<R$ for some $R>0$. If $z=a$ is not removable singularity of $f$, show that $e^{f(z)}$ has an essential singularity at $z=a$. Since it's not removable ...
2
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1answer
79 views

How to describe formal functions around the node of an algebraic curve

Let $C$ be a smooth proper curve over an algebraically closed field $k$ with $p,q \in C$ two distinct points. Let $X = C / p \sim q$ be the nodal curve obtained by gluing $p$ and $q$. Denote the node ...
0
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1answer
124 views

Show that $f$ is a polynomial of degree $m$

Show that if an entire function $f$ has a pole at $\infty$ of order $m$ then it is a polynomial of degree $m$. My try:$f$ has a pole of order $m\implies g(z)=f(\frac{1}{z}) $ has a pole of order $m$...
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0answers
45 views

Find the metric around a trefoil singularity

Singular surfaces in $\mathbb{C}^2$ can be described by knots or links: $z_1^2-z_2^2=0$ is the Hopf link, $z_1^2-z_2^3=0$ is the trefoil, etc. I want to know how to determine the metric on the surface ...
2
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1answer
174 views

Intuitively, why does there need to be an exceptional divisor?

This question is a bit vague, but I hope its still good enough for this site. When resolving a singularity of, say, a curve by blowing up a point, we get new variety that besides the strict transform ...
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0answers
26 views

Specify point singularities

I have problem with point singularities at $\infty$. For example: Determine the nature of the point at infinity functions. For zero or pole enter times. we have function $f(z)=\frac{z-\frac{\pi}{2}}{...
2
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1answer
100 views

How to make this integrand nonsingular?

$$T = 4\sqrt{\frac{l}{2g}} \int_0^x{\frac{dy}{\sqrt{\cos(y)-\cos(x)}}}$$ So do i just have to find a way so that the denominator does not result in 0? I am to later evaluate it with x = 51. Which ...
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1answer
350 views

How can I prove that a singularity is removable, a pole, etc.?

Let's sy I have something like $f(z)=\frac{z^6+1}{z^2+1}$. I know that the singularity is at $z=\pm i$ and that it ends up being a $\frac{0}{0}$ case, but is that an indication for what type of ...
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1answer
469 views

Singularities at infinity and laurent series

For this problem, I have three questions: 1. Why do we calculate the residue at the infinity? Is it because all the four poles are in contained in the disk? 2. When calculating $f(1/z)$, don't we just ...
3
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2answers
41 views

How can I see mathematically that these two singularities are different?

I came across these two curves while reading the wikipedia page on singularity theory: $$ y^2=x^3+x^2 $$ and $$ y^2=x^3 $$ The page says the cusp at $(0,0)$ can be seen to be qualitatively different ...
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0answers
41 views

Gloabal and Local data on Singular Varieties

Suppose we have a subvariety (codimension one) inside a smooth variety with a fixed divisor class D. However, let it be a singular one. I'm just wondering how much the usual techniques (such as ...
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0answers
50 views

How can one define singular foliations in terms of charts?

Let $\mathcal F$ be a (regular) codimension $q$ foliation. A foliated chart is an open set $U$ and a local diffeomorphism $\varphi:U\to R^{q}\times R^{n-q}$, where $\varphi^1=c^1,\dots,\varphi^q=c^q$ ...