# Tagged Questions

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: http://en.m....

989 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 ...
149 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 ?
68 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 ...
139 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 ...
16 views

### Versal deformation of $x^3+y^3$

I am trying to compute fundamental group of complement to discriminant hypersurface of $f=x^3+y^3$ singularity via Zarisski-van Kampen theorem. So, I need a versal deformation of singularity to ...
96 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 ...
143 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 ...
41 views

51 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, ...
38 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 $∂p/∂x$ ...
100 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 ...
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 ...
129 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 ...
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 $\operatorname{mult}(f,P) = \max\{n\in\mathbb{N}\mid f\in\mathfrak{m}_P^{n}\}$, ...
62 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 https://www....
54 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 ...
76 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 ...
64 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 ...
44 views

### A complement of Milnor's book on singularities with exercises/examples

I'm currently reading "Singular Point of Complex Hypersurfaces" by Milnor. This is really a great book, but I did realize I didn't saw any concrete examples, computations etc ... I was wondering if ...
44 views

### Projective and affine varieties: differences, advantages and why two definitions

I have recently started to learn algebraic geometry and this question has been bugging me. An affine variety is a zero set of a collection of polynomials in affine space and a projective variety is ...
28 views

### How to prove the minimal resolution of double rational points can be achieved by iterated blow-ups

In the Slodowy's survey on Kleinian singularities, there is a statement that the minimal resolution of Kleinian singularities can be obtained by iterated blow-ups, I want to know the detail of the ...
24 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 $a, b \in S^n$ ...
43 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 ...
33 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 ...
38 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 ...
53 views

57 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 ...
70 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 $Y$?
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 ...
112 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 ...
29 views

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 f'(... 0answers 142 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; $$K_X=\left[\begin{matrix}1 &-\frac12 &-\frac12 \\ -\frac12 &1 &-\frac12 \\ -\frac12 &-\... 0answers 71 views ### Do there exist double points on a surface in \mathbb{P}_{\mathbb{C}}^3 that are not rational? The title explains it all. I'm familiar with the du val singularities on surfaces, also apparently known as rational double points (http://en.wikipedia.org/wiki/Du_Val_singularity). In http://... 0answers 73 views ### A singular boundary value problem Is there any numerical approach to solve a BVPs for ODEs of the form: y'=\frac{ky^2-y^{3/2}-y}{\beta t} with initial point (0,y0)? I know a problem of the form y' = \frac{S}{t}y+f(t,y) with ... 0answers 83 views ### How to analyze the asymptotic properties of this function? Let the function$$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where \mathbf{k} ,\mathbf{r}\in\mathbb{R}^2, and \Omega \subset \mathbb{R}^2 is some finite region ... 0answers 44 views ### What's a algebraically isolated singularirty? What's its formal definition and how does it differ from a "simple" isolated singularity? I'm thinking about integrable 1-forms here, though the definition may be more general. 0answers 88 views ### Topology of singularities In the theory of surface singularities, it is well known that the topology of singularities is determined by its resolution dual graph. What`s the meaning of the topology of singularities? Here the ... 0answers 127 views ### Question on definition of 3-fold cDV singularities A 3-fold compound Du Val singularity of type A_{k} \ (D_{k}, E_{k}) is defined as a singular point on a 3-fold which is locally given by$$ F(x,y,z,w)=f(x,y,z)+wg(x,y,z,w)=0, $$where g(x,y,z,w) ... 0answers 27 views ### Proof that if z_0 is a zero of order k of f then \exists analytic g saisfying this expression Let f:S\to \mathbb{C} be analytic, z_0 \in S be of order k of f. Prove that there exists an analytic function g satisfying$$\frac{f'(z)}{f(z)}=\frac{k}{z-z_0}+g(z)$$\forall z in \... 0answers 38 views ### Blowing up to get normal crossings Let X \subset W be an algebraic projective set, whose irreducible components are smooth, and W is some ambient smooth variety (everything over \mathbb C). Even if the irreducible components of ... 0answers 44 views ### What is the image near the essential singularity z=0 of \cos(1/z)? Determine the image near the essential singularity z=0 of the function \cos(1/z). i.e. if f(z)=\cos(1/z), What is f\left(\mathcal{B}_{\varepsilon}(0)\right) for \varepsilon > 0? Remark: ... 0answers 35 views ### Difficulty with Milnor number I am reading through the wikipedia page of Milnor number : https://en.wikipedia.org/wiki/Milnor_number#Examples I am reading example 2 where they calculate the Milnor number of f(x,y)=x^3+xy^2. So ... 0answers 18 views ### p a nonsingular point in V and V(f)\cap T_p(V) then f|_{T_p(V)} has a factor of multiplicity \geq 2. This is a problem from Ideals, Varieties, and Algorithms by Cox et. at. Let V \subset k^n be a hypersurface with I(V ) = \langle f \rangle. Show that if V is not a hyperplane and p \in V ... 0answers 37 views ### Local ring in singularity theory and algebraic geometry In the book Singularity theory 1 by Arnold et al., page 13, there is a definition of a local algebra of singularity, which I tried and failed to rephrase in the language of algebraic geometry. It is ... 0answers 22 views ### How to numerically solve equation with regular singularity problem? Good day, Here is my equation$$ (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],$q_2(x,\Omega)$[... 0answers 59 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}} \...
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 ...