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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35
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949 views

What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?

I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
5
votes
0answers
141 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 ?
4
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0answers
67 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
votes
0answers
132 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 ...
3
votes
0answers
91 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 ...
3
votes
0answers
136 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 ...
2
votes
0answers
64 views

Canonical scheme structure on the singular locus of a variety

Let $X$ be subvariety of affine space $\mathbb{A}_{k}^n$, where $k$ is a field, and suppose $X$ is given by equations $$X:(F_1=\cdots = F_m=0)\subset \mathbb{A}^n.$$ Then $X$ is singular at $p\in ...
2
votes
0answers
39 views

Integers characterizing singularities of algebraic curves

The problem in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field ...
2
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0answers
36 views

Understanding singularities through Jacobian

In kinematics of mechanisms we derive the constraint equations depending on the architecture and then analyse the singularities of the mechanisms by deriving their Jacobian matrices. For example: ...
2
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0answers
46 views

How to determine the type of singularities in affine varieties

Let $X$ be the affine variety in $\mathbb C^3$ (coordinates $a, b, c$) defined as the zero set of $ac - b^2$. The variety has a "double point singularity" at the origin. This is somewhat intuitive, ...
2
votes
0answers
32 views

PDE Singularity and time of shock

Given the following PDE: $$∂p/∂t+p*∂p/∂x=0$$ $$p=p(x,t)$$ $$p(x,0)=e^{-x^2}$$ It is requested to find its solution, show that it is not well behaved by verifying what happens to $∂p/∂t$ and $∂p/∂x$ ...
2
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0answers
97 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 ...
2
votes
0answers
52 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 ...
2
votes
0answers
119 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 ...
2
votes
0answers
67 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 $\operatorname{mult}(f,P) = \max\{n\in\mathbb{N}\mid f\in\mathfrak{m}_P^{n}\}$, ...
2
votes
0answers
61 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 ...
2
votes
0answers
53 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 ...
2
votes
0answers
72 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 ...
2
votes
0answers
62 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 ...
1
vote
0answers
22 views

Simple singularities of a vector field

Let $M^n \subset \mathbb{R}^{n+1}$ a compact and orientable hypersurface of even dimension, with Gauss map $\gamma : M \to S^n$, that is, $\gamma(p)$ is normal to $M$ at $p \in M$. Let $a, b \in S^n$ ...
1
vote
0answers
36 views

Compute the genus of a curve with a flex point

The genus of a smooth plane curve is $g=\frac{(d-1)(d-2)}{2}$ and I know that if the curve has $n$ nodes the genus decreases by $n$. What happens if the curve has singular (non ordinary) points? In ...
1
vote
0answers
32 views

The set of curves of degree $d$ with two singular points or a degenerate singular point is closed?

Suppose $d>0$, $k$ is a field and $L_d$ is the space of projective plane curves over $k$ of degree $d$, namely $L_d=\mathbb P(k[X,Y,Z]_d)$, where $k[X,Y,Z]_d$ is the vector space of homogeneous ...
1
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0answers
33 views

Gimbal lock easier to control with quaternions?

Using quaternions doesn't resolve the issue of gimbal lock, but make is more controllable... how come? They use less memory, and are commutable, and provide an smooth rotation along nonlinear ...
1
vote
0answers
36 views

On the dog-bone contour around [-1,1], what are the arguments of these two lines approaching the real axis from above and below?

I am using a dog-bone contour to integrate around the interval [-1,1]. (-1 and +1 are branch points of the integrand.) I am using the principal branch of log, so I am restricting its argument to ...
1
vote
0answers
47 views

How to solve a singularity problem for diferential equation?

Good day, Here is my equation $$ (x-1)^2 (x-h)^2q_1(x,w)y''(x)+(x-1)(x-h)q_2(x,w)y'(x)+q_3(x,w)y(x)=0 $$ (with $q_1(x,w), q_2(x,w), q_3(x,w)$ regular functions). The boundary conditions are $$ ...
1
vote
0answers
21 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 ...
1
vote
0answers
51 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 ...
1
vote
0answers
58 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 $Y$?
1
vote
0answers
19 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 ...
1
vote
0answers
89 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 ...
1
vote
0answers
29 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 ...
1
vote
0answers
134 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 ...
1
vote
0answers
67 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 ...
1
vote
0answers
72 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 ...
1
vote
0answers
83 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 ...
1
vote
0answers
44 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.
1
vote
0answers
86 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 ...
1
vote
0answers
120 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 $g(x,y,z,w)$ ...
0
votes
0answers
10 views

divergence theorem with singularity at r = 0

I am trying to evaluate the volume integral given by \begin{align} \int_V [\nabla(\vec{x} \cdot \vec{u}) - \nabla \cdot (\vec{x}\vec{a})] dV \end{align} where $\vec{x}$ is the position vector and ...
0
votes
0answers
35 views

Blowing up to get normal crossings

Let $X \subset W$ be an algebraic projective set, whose irreducible components are smooth, and $W$ is some ambient smooth variety (everything over $\mathbb C$). Even if the irreducible components of ...
0
votes
0answers
43 views

What is the image near the essential singularity $z=0$ of $\cos(1/z)$?

Determine the image near the essential singularity $z=0$ of the function $\cos(1/z)$. i.e. if $f(z)=\cos(1/z)$, What is $f\left(\mathcal{B}_{\varepsilon}(0)\right)$ for $\varepsilon > 0$? Remark: ...
0
votes
0answers
28 views

Difficulty with Milnor number

I am reading through the wikipedia page of Milnor number : https://en.wikipedia.org/wiki/Milnor_number#Examples I am reading example 2 where they calculate the Milnor number of $f(x,y)=x^3+xy^2$. So ...
0
votes
0answers
18 views

$p$ a nonsingular point in $V$ and $V(f)\cap T_p(V)$ then $f|_{T_p(V)}$ has a factor of multiplicity $\geq 2$.

This is a problem from Ideals, Varieties, and Algorithms by Cox et. at. Let $V \subset k^n$ be a hypersurface with $I(V ) = \langle f \rangle$. Show that if $V$ is not a hyperplane and $p \in V$ ...
0
votes
0answers
30 views

Local ring in singularity theory and algebraic geometry

In the book Singularity theory 1 by Arnold et al., page 13, there is a definition of a local algebra of singularity, which I tried and failed to rephrase in the language of algebraic geometry. It is ...
0
votes
0answers
18 views

How to numerically solve equation with regular singularity problem?

Good day, Here is my equation $$ (x-1)^2 (x-h)^2q_1(x,\Omega)y''(x)+(x-1)(x-h)q_2(x,\Omega)y'(x)+q_3(x,\Omega)y(x)=0 $$ ( with $q_1(x,\Omega)$[polynomial function 38 degree], $q_2(x,\Omega)$ ...
0
votes
0answers
58 views

How to get asymptotic form of the integrals with special functions?

I got difficulty when I try to plot I(x) for $m=1$ and $t=0.2$. The questions is how to get the asymptotic form of the following integral? $I(x,t)=\int_{0}^{\infty} \frac{f(y)}{2 \sqrt{\pi t}} ...
0
votes
0answers
17 views

Identification of pole

I would like to know whether the following statement is true or not If $f$ is an analytic function satisfied $|f(z)|\to\infty$ when $z\to z_0$, then $f$ is a pole at $z=z_0$ I would say the ...
0
votes
0answers
17 views

How to understand multifractal and multifractal spectrum?

A case I encountered with to illustrate multifractal is to assign different subintervals of Cantor set with different mass. In detail for each time, the interval is divided into 3 equal-length parts ...
0
votes
0answers
42 views

Is the complement of the singular locus of an algebraic variety a topological manifold?

Could someone explain to me clearly, please, why is the complement of the singular locus of an algebraic variety a topological manifold ? A lot of thanks for your help.
0
votes
0answers
39 views

About singularity

Let $X$ be a normal affine variety over $\mathbb{C}$. Q1. Let $x\in X$ be a singularity with a Cartier divisor $x\in D$. Then $\mathcal{O}_{X,x}$ is Cohen-Macaulay if and only if $\mathcal{O}_{D,x}$ ...