In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. (Def: ...

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103 views

Singularities on a weighted projective curve

Let $C$ be the curve of degree $3$ defined over $\mathbb{C}$ given by $$x(y+z)=y^3-z^3$$ which lives in the weighed projective space $\mathbb{P}(x,y,z)=\mathbb{P}(2,1,1)$. Is the curve singular ?
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1answer
85 views

Laurent series of an analytic function divided by $z$

This is a probably basic question about Laurent series. Say $g(z)$ is an analytic function, that $g(0) = 0$, and $f(z) = g(z)/z$. My textbook says $z = 0$ is a removable singularity of $f(z)$. A ...
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1answer
58 views

Question regarding singularity of a complex function

Consider the function $$f(z) = {1 \over (z-i)(z+i)}$$ with a Laurent series expansion at $z_0=i$ on a domain $\;\Omega=\left\{z\in \mathbb{C}:2\lt\left|z-i\right|\right\}$ $$\begin{eqnarray}f(z)={1 ...
2
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1answer
139 views

Removable singularities of a holomorphic function

So, I'm a little confused about removable singularities. Consider the function below: $$f(z)=\frac{1}{(1+z^2)^{2/3}}$$ Obviously, we have isolated singularities at the points $z = \pm i$. ...
2
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1answer
116 views

Singularities in (Elementary) Real Algebraic Geometry

I've taken an introductory course in algebraic geometry, and am currently studying Sumio Watanabe's book, "Algebraic Geometry and Statistical Learning Theory". In this book Watanabe gives a, possibly ...
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1answer
46 views

singularity and degeneracy of an ODE

I have trouble distinguishing the difference of singularity and degeneracy in the context of ODE theory. Could anyone give me a couple of examples in illustrating the difference of singular point and ...
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1answer
91 views

Slodowy slices for non ADE type Lie algebras

In the first answer to: http://mathoverflow.net/questions/16026/the-finite-subgroups-of-sl2-c the correspondence between transverse slices of subregular elements of the nilpotent cone for a simple Lie ...
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1answer
27 views

Singularity of Product of two complex function $f$ and $g$

Suppose $f$ has an essential Singularity at $z = a$ and $g$ has a pole at $z = a$. Then the product $fg$ has an essential Singularity at $z = a $. Is this hold if $g$ has removable ...
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28 views

Singular values of rectangular matrix

can any one explain me the need for singular values of a matrix. Explanation with a practical example will be appreciated
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64 views

Minimal resolutions of singularities in higher dimensions and their cobordism class.

For a given singular algebraic variety over $\mathbb{C}$, how many minimal resolutions can exist? More specifically, consider a projective quadratic cone $Q$ in $\mathbb{C}P^{4}$ (with homogeneous ...
4
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117 views

Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
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87 views

Manifolds, isolated singularities, open map

I can't figure out how to solve the following question: 1) Let $M$ and $N$ be $C^k$-manifolds, such that $\dim M = \dim N = n>1$, and let $f:M\rightarrow N$ be a $C^k$-function. Show that if the ...
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48 views

Requesting information on constructed discontinuous functions (from any perspective)

Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Define the function $F:\mathbb{R}\rightarrow \mathbb{R}$ as $$F(x)=f(x)\prod_{n=1}^\infty\frac{x-\frac{1}{n}}{x-\frac{1}{n}}$$ I ...
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85 views

Could this be called Renormalization?

Quoted from   Space-Time Approach to Quantum Electrodynamics   by R. P. Feynman, Phys. Rev. 76, 769 1949 : We desire to make a modification of quantum electrodynamics analogous to the ...
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76 views

necessary and sufficient conditions for having an irreducible hyperplane section

Let $S \subset \mathbb{P}^3$ be a surface different to a cone. If $S$ is a smooth variety, then its generic hyperplane section is irreducible. I can imagine a similar result holds for a surface with ...
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57 views

Relation of two definitions of 'Singular Point'

In my lecture we defined: For $(a,b)=P\in V(f)$, $P$ is a singular point if $\operatorname{mult}(f,P)>1$, where $\operatorname{mult}(f,P) = \max\{n\in\mathbb{N}\mid f\in\mathfrak{m}_P^{n}\}$, ...
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58 views

The multiplicity of $X$ at $x$ does not change when $X$ is cut by a generic hypersurface: what are those generic conditions?

Given an algebraic variety $X$ with a point $x \in X$, the multiplicity of $X$ at $x$ is defined as the multiplicity of the maximal ideal of $x$ in the local ring $\mathcal{O}_{X,x}$. In ...
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107 views

Puiseux series and Resolution of Singularities

I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities. So the Newton's method gives us a Puiseux series parametrizing the branches of ...
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0answers
46 views

I have an infinite solution to an ODE even though it has only a regular singular point

I have the ODE: $\displaystyle y''(x)+\frac{y'(x)}{x+1}+y(x)=0$ I know that this has a regular singular point at $x=-1$, as $(1+x)^{-1}$ has only a first order pole, and $1$ has no pole at all, and ...
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56 views

Algorithm for checking that a singularity is an ordinary double point

Let a complete intersection $d$-fold singularity be cut out by $k$ polynomials $\{f_1,\ldots,f_k\}$ in $\mathbb{C}^{k+d}$. Is there a relatively simple algorithm to check whether this is analytically ...
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58 views

How do I check whether an orbifold admits deformations?

Orbifolds $\mathbb{C}^2/\mathbb{Z}_n$, given by the action $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$ with $\zeta$ a primitive $n^\text{th}$ root of unity, admit smooth deformations. This is because ...
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14 views

Eliminate singularity from a PDE system

I need some help with the theory of PDE systems, which I am not very familiar with. Assume $f$ and $g$ are two functions of $x,y$ in the real plane. I have the equation ...
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29 views

magma: singularity type (IsNode?)

Does magma know the singularity type of a point in a scheme? I've seen some functions but they all seem to be about curves. This is my context: I have a surface in $P^2\times P^1$ with zero ...
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49 views

Is exceptional divisor always a Projective bundle over the centre?

Let $f:\tilde X \rightarrow X$ be a blow up at center Z. Is $f^{-1}(z)=\mathbb{P}^{k}$ ?for some $k$, $\forall$ $z\in Z$ In the case of blow-up of $\mathbb{A}^{n}$ at origin it is very clear that the ...
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28 views

Residue Theorem: can I say this?

I know that $f(z)$ is analitic everywhere except of $z=0$, where it has a pole of order $k$. Can I say that $f'(z)$ will have a pole of order $k+1$? For example, if $f(z)=\frac{1}{z^k} \rightarrow ...
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69 views

If Gaussian random vector has singular covariance matrix, isn't there probability density function?

I got a complicated problem. Suppose that Gaussian random vector having Covariance matrix; $$K_X=\left[\begin{matrix}1 &-\frac12 &-\frac12 \\ -\frac12 &1 &-\frac12 \\ -\frac12 ...
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66 views

Do there exist double points on a surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?

The title explains it all. I'm familiar with the du val singularities on surfaces, also apparently known as rational double points (http://en.wikipedia.org/wiki/Du_Val_singularity). In ...
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67 views

A singular boundary value problem

Is there any numerical approach to solve a BVPs for ODEs of the form: $y'=\frac{ky^2-y^{3/2}-y}{\beta t}$ with initial point $(0,y0)$? I know a problem of the form $y' = \frac{S}{t}y+f(t,y)$ with ...
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74 views

How to analyze the asymptotic properties of this function?

Let the function $$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where $\mathbf{k} ,\mathbf{r}\in\mathbb{R}^2$, and $\Omega \subset \mathbb{R}^2$ is some finite region ...
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43 views

What's a algebraically isolated singularirty?

What's its formal definition and how does it differ from a "simple" isolated singularity? I'm thinking about integrable $1$-forms here, though the definition may be more general.
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76 views

Topology of singularities

In the theory of surface singularities, it is well known that the topology of singularities is determined by its resolution dual graph. What`s the meaning of the topology of singularities? Here the ...
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0answers
99 views

Question on definition of 3-fold cDV singularities

A 3-fold compound Du Val singularity of type $A_{k} \ (D_{k}, E_{k})$ is defined as a singular point on a 3-fold which is locally given by $$ F(x,y,z,w)=f(x,y,z)+wg(x,y,z,w)=0, $$ where $g(x,y,z,w)$ ...
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18 views

Second Intrinsic Derivative

I'm trying to understand the second intrinsic derivative but I just don't get it. For example for an $A_r$ singularity $y_{1}=t_{1}, y_{2}=t_{2}, y_{r-1}=t_{r-1}, \cdots , y_{r}=t_1x + t_2x^2+\cdots ...
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61 views

resolution of a surface singularity

Let $f: Y \to (0\in X)$ be a resolution of a surface singularity and $R^1f_*\mathcal{O}_Y=0$. Then $H^1(Y,\mathcal{O}_Y)=0$. Why? Is it clear? How can I show it?
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24 views

Singular values and trace?

Given that $X$ and $Y$ are positive definite matrices, how can I bound the singular values $\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?
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10 views

question on singularity theory and normal forms of local function germs

Hello I realize this might be a difficult topic but I was just trying to practice for my advanced class on singularities I wanted to compute normal forms for corank 2 singularities of the function ...
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23 views

Laurent series about which point?

I have been given the question: "Find the laurent series of $1/(1-z^3)$". No other information. The question was posed in the process of finding the series' residual and in the answer I can see that ...
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0answers
15 views

Quadratic singularities and local curves.

Everything is to be understood over the complex field. Assume you have two finite dimensional $\mathbb{C}$-vectorial space $V$ and $W$. You are given a bilinear form : $$\phi:V\times V\rightarrow W ...
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28 views

Singularities and Residues [Demonstration]

How could I solve the following problem: "Consider $F$, an analytic function, so that $$f(z)=F(\frac{1}{z-1})$$ has a pole. Demonstrate that F(z) is a polynomial." I know that an analytic function ...
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46 views

Du Val singularities, Reid‘s D$_4$

Maybe this is a stupid question, but I feel puzzled. In Reid's note: Chapters on algebraic surfaces, in 4.2 Du Val singularities, example D$_4$, he writes the equation of D$_4$: x$^2$+y$^3$+z$^3$=0, ...
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17 views

$a$ is a non isolated singularity of $f (z)$

If a point $a$ is a non-isolated singularity of a function $f(z)$ and $|f(z)|$ is bounded in some neighbourhood of $a$ then what kind of singularity of $f(z)$ occurs at $a$? Removable singularity ...
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20 views

show that a function on annulus has a removable singularity

How can I show, that for $f \in \mathcal{O} \cap L_2(U \setminus \{z_0\})$, f has a removable singularity at $z_0$? I should use this fact: if $$f = \sum_{n= - \infty} ^{\infty} a_n (z-z_0)^n \in ...
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21 views

How to determine singularities of a series?

Given a double Fourier series, how do we determine its singularities ? PS: I wonder how we find singularities(mathematically) if a function cannot be expressed in a closed form.
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38 views

Term for Multiple Functions that Share Critical Points?

Is there a term for when multiple functions share each other's critical points? Or, in general, when one function has a subset of the critical points of another?
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94 views

Cusp singularity

What is the most general way of defining cusps singularities for a continuous curve $\mathbf{r}:(a,b)\to\mathbb{R}^2$ and how can we define them? What are the minimum requirements for the curve ...
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40 views

Characteristics of Singularity Points

Determine the character of the singularity at $z=0$ for each of the following functions: \begin{align} &\frac{1/z^{7}}{e^{z}-1} \tag{1} \\ &\frac{}{} \\ ...
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31 views

The normalisation map is a bi-Lipschitz map?

Let $X$ a reduced analytic space, $n: W \rightarrow X$ the normalisation map, $W$ the normalisation of $X$ and $S$ the singular set of $X$. When we restrict $n$ to $W\setminus n^{-1}(S)$, we know that ...
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65 views

Alternative Proof why algebraic groups are smooth

Definitions: Let $k$ be a field, by an affine algebraic group we will mean an affine group scheme $G$ over $Spec(k)$ with the property that the coordinate ring of the underlying affine scheme ...
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96 views

Stratification of a smooth map

I am trying to do an exercise. Namely, find the Thom-Boardman stratification of the smooth map $f(x,y,a,b,c,d)=x^2y+y^3+a(x^2+y^2)+bx+cy$, where $a,b,c$ are parameters. As I have seen, this is also ...