# Questions tagged [singularity-theory]

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238 questions
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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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. $$F(r) = \... 1answer 53 views ### What type of singularity is z=0 Find the residue of \frac {\cot(z)\coth(z)}{z^3} at z=0. In my notebook, i expanded the functions and took the coefficient of 1/z as residue, which is -7/45 I couldn't recall what type of ... 0answers 80 views ### Why doesn't this point of the quintic have a singular Hessian? I am considering the quintic:$$ 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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... 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 ... 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 ... 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}}{... 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)}}}$$So do i just have to find a way so that the denominator does not result in 0? I am to later evaluate it with x = 51. Which ... 1answer 350 views ### How can I prove that a singularity is removable, a pole, etc.? Let's sy I have something like f(z)=\frac{z^6+1}{z^2+1}. I know that the singularity is at z=\pm i and that it ends up being a \frac{0}{0} case, but is that an indication for what type of ... 1answer 469 views ### Singularities at infinity and laurent series For this problem, I have three questions: 1. Why do we calculate the residue at the infinity? Is it because all the four poles are in contained in the disk? 2. When calculating f(1/z), don't we just ... 2answers 41 views ### How can I see mathematically that these two singularities are different? I came across these two curves while reading the wikipedia page on singularity theory:$$ 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 ...
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 $\varphi^1=c^1,\dots,\varphi^q=c^q$ ...