Questions tagged [singularity-theory]

This tag is for questions relating to Singularity Theory. In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes.

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Check if singularity of curve is node in positive characteristic

Given a plane curve it is easy to check whether a point is singular by using the Jacobi criterion. However, I am stuck with checking whether it is a node or worse, especially in the case of positive ...
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Area near the singularities

Given a complex meromorphic function $f$ with only simple poles and $|f'|\neq0$ everywhere, we furthermore have the condition that $$\int_{\mathbb{R}^2}\frac{|f'|^2}{(1+|f|^2)^2}\text{d}x\text{d}y<\...
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Resolution of ADE- singularities: The second blow-up for the A_2 singularity

I try to resolve the surface singularity $X\subset \mathbb{A}^3$ of type $A_2$. It is defined by $x^2+y^2+z^3=0$. I expect to need at least two blow-ups. But after the first blow-up of $X$ the strict ...
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A smooth map that is singular everywhere

Let $U$ be an open subset of $\mathbb{R}^d$, and let $F \colon U \to \mathbb{R}^{d +c}$ be a smooth map whose rank is everywhere equal to $d-1$: $$\mathrm{rank} (dF_{p}) = d-1 \quad \text{for all $p \...
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The commutator map

I am trying to study the commutator map of a given Lie group $G$: $$\mu : G\times G\to G,\ \mu(x,y)=[x,y]=xyx^{-1}y^{-1}$$ I am interested in: Its singular points (where the the differential is not ...
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From differential form to vector fields

In page 13 of this course https://www.sissa.it/fa/download/publications/remizov.pdf from the equation $$ (F_x+pF_y) dx + F_pdp=0 \qquad (*) $$ I do not understand how we can get $$ \dot x =F_p, \quad ...
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singularity and blow up for the cusp

Take the exercise from Arnold's book at page 10 where we are told to solve the singularity at $0$ of the curve $x^2=y^3$. The solution is given by the following graphs : From the first graph, we ...
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Singularity of solutions of a differential equation as functions of its eigenvalue

This may be a naive question and I apologize if it's not stated concisely and rigorously. I try to pose it first, and then explain my thoughts and comments on it; so please bear with me. Let $\...
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Cup product of cohomology.

Suppose $[K \colon {\Bbb Q}_p] < \infty$ and that $\mu_p \in K$. We shall consider the cup product $$ H^1(G, {\Bbb Z}/p) \times H^1(G, {\mu_p}) \overset{\cup}{\to} H^2(G, \mu_p) \cong {\Bbb Z}/p{\...
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Are these the isolated singularities?

Find the isolated singularities of the function $$f(z)=\frac{e^z-1}{z^4(z^2+4\pi^2)}$$ $$$$ I have done the following : We set the denominator equal to zero : $$z^4(z^2+4\pi^2)=0 \Rightarrow z=0 \text{...
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When does a singular foliation admit a resolution?

For example, Any holomorphic singular foliations on a complex manifold X of dimension n locally admits a finite resolution by finitely generated free OX-modules. On the other hand, there are more ...
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The algebraic implication of not being a Noetherian algebra

I have a doubt as to what algebraic implication it has that a ring is not noetherian. I have considered the ring of germs at the origin \begin{equation} \varepsilon_{n}=\lbrace f\colon(\mathbb{R}^{n},...
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Examples of real High Milnor Du Val Quartics

I am looking for examples of specific quartic projective hypersurfaces over $\mathbb{P}^{3}$. So I am going off the fact the famous Kummer surface, under some parameters, have 16 real $A_{1}$ ...
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Book of applications of the theory of singularities to the geometry of surfaces in $\mathbb{R}^3$

I've been searching the web for the book : BANCHOFF, T., GAFFNEY, T. and MCCRORY, C. Cusps of Gauss mappings (Research Notes in Mathematics 55, Pitman, 1981) on the web but I have not found references,...
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Fourier-transform of a delta-function of a square of the space-time interval

What is the general framework for calculating expressions such as $$ I = \int d^4x e^{-i(px)} \delta(x^2) $$ where $x^2=x_0^2-\vec{x}^2$? The problem here is that the delta-function carves out not a ...
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Why classify all singularities? Application of classification

I was given a book about classifying all singularities of a complex hypersurface. What is the importance of classifying all singularities? In which areas do singularities pop up and how does ...
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Why is the monodromy of a fibration over $\mathbb{D}$ whose fibre has boundary trivial near the boundary?

I am reading Arnol'd's book "Singularities of Differentiable Maps, Volume 2", and I'm trying to understand geometric monodromy, as he defines it on page 10. I have simplified the assumptions ...
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Contraction of loops on algebraic surfaces.

Suppose ${\Bbb C}$ be a complex number field and $S$ be a projective smooth surface over ${\Bbb C}$. We consider a finite etale Galois covering $\pi \colon T \to S$ of degree $d$. We consider the ...
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Birational Map-codimension

Let $X$ be an algebraic variety over an algebraically closed field of characteristic zero. Assume that $X$ is embedded as a closed subvariety in $\mathbb{P}^{n}.$ Is it possible that there is a proper ...
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2 answers
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Simple doubt on singularities and poles theory.

I found the following definition online: A point $z_0$ is called a pole of order $m$ of $f(z)$ if $1/f$ has a zero of order $m$ at $z_0$. I was wondering if this definition is valid at every case, and ...
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Characterizing Distributions as sum of a regular distribution and $\delta^\alpha$ distributions?

Please mark this as duplicate if this has been asked before. I started a deep dive in distribution theory on $\mathbb{R}^n$ I want to understand the singular support better. I know that there are ...
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Resolution of singularities of analytic spaces

It seems to me that the following resolution of singularities theorem (or a modification) is known to specialists but I have trouble finding references. Let $X$ be a complex analytic space, then there ...
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110 views

Relationship between exact symplectomorphisms and Hamiltonian diffeomorphisms (in the context of Milnor fibers)

I asked this same question on Math Overflow but I might get a quicker response if I post it here as well. Here is a preamble. Suppose we have a symplectic manifold $(M,d\lambda)$ with an exact ...
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Show that it is not a Notherian module

Let the ring $\varepsilon_n=\lbrace f\colon (\mathbb{R}^{n},0)\longrightarrow \mathbb{R}: f \quad\textit{is map germ}\rbrace$ and maximal ideal $\mathcal{M}_{n}=\lbrace f\in\varepsilon_{n}:[f]_{0}=0\...
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2 votes
0 answers
63 views

calculate the kernel of a germ map

I have a question, more about how to calculate the kernel of a certain map: Let the ring $\varepsilon_n=\lbrace f\colon (\mathbb{R}^{n},0)\longrightarrow \mathbb{R}: f \quad\textit{is map germ}\...
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1 answer
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Connectedness of exceptional divisors

Let $X$ be a quasi-projective variety over $\mathbb{C}$. Let $I$ be an ideal sheaf supported at a closed point on $X$. Is the exceptional divisor for the blow-up of $X$ along the ideal sheaf $I$, ...
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The germ diffeomorphism set is a group

Definition. Let germ $\phi\colon (\mathbb{R}^n,0)\to (\mathbb{R}^n,0)$, is germ of diffeomorphism, if there exist $\psi\colon (\mathbb{R}^n,0)\to (\mathbb{R}^{n},0)$, such that $\psi\circ \phi = \...
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A criterion for surjectivety in terms of rank of the differential

Let $M$ and $N$ be smooth compact $3$-dimensional manifolds. Assume that $f \colon M \to N$ is a $C^2$-map, such that $\operatorname{rk} \ Df = 3$ in at least one point; if $\operatorname{rk} Df \neq ...
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4 votes
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Derivatives of a recursively and implicitly defined polynomial

I'm studying Frobenius Manifolds associated with $A_n$-type singularities and in order to prove some results about their potentials I need to calculate the following thing. Assume that $n$ is a fixed ...
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J.-P. Serre, Algebraic groups and class fields, IV Proposition 7: Why is $\beta = g \omega$?

Suppose $(S',Q)$ is the germ of a singular algebraic curve curve with normalisation $S \to S'$. Let $\mathcal O_Q'$ be the local ring of $S'$ at $Q$, let $\mathcal O_Q$ be its normalization, and let $\...
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1 vote
1 answer
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Why does the conductor ideal contain a power of the radical?

Let $X'$ be a singular algebraic curve with singularity $Q \in X'$ and normalization $X \to X'$. Suppose $\mathcal O_Q'$ is the local ring of $X'$ at $Q$ and $\mathcal O_Q$ is its integral closure in ...
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66 views

Asymptotic directions on a parabolic set

Let $\gamma\colon I\to S$, where $I$ is an interval and $S$ is a surface then the curvature restricted to $\gamma$ is $K(\gamma(t)))=0$ in the parabolic set $\gamma$, if we derive the \begin{equation*}...
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1 vote
0 answers
127 views

Canonical morphism of dualizing sheaves under normalization.

The following appears in a paper by Hwang and Oguiso. Let $V'$ be a complex variety with the property that its normalization is smooth and the dualizing sheaf $ω_{V'}$ is invertible. Denoting by $ν: ...
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What is "log canonical threshold" in simple terms?

I'm currently learning some basic algebraic geometry and am interested in singularities of varieties. In reading about classifying singular points I've come across the log canonical threshold (LCT) as ...
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3 votes
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63 views

ADE singularity and Milnor lattice

I am reading this paper: https://arxiv.org/abs/0810.2687 by A. J. de Jong and Robert Friedman. In the proof of Theorem 4.10, a singularity of the following type shows up $$y^2=x^3+z^{6d-1}.$$ When $d=...
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1 vote
0 answers
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Example of a real analytic subset which is not subanalytic?

It is mentioned on page 40 of Shiota's book, "geometry of subanalytic and semialgebraic sets", that a (real) analytic set in $\mathbb{R}^n$ is not necessarily subanalytic. Here a set $S \...
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Classifying Singular Points as Regular/Irregular for Differential Equation

The question asks to find the singular points of the Differential Equation and to classify each of them as regular/irregular: $x(x^2+1)^2y''+y=0$ Edit: I gave it another shot and I ended up with 3 ...
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4 votes
1 answer
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Derivations of the algebra $K[x,y]/(y^2-x^3)$

Let $K$ be a field of characteristic zero, and let $S=K[x,y]/(y^2-x^3)$. It is easy to see that $S\cong R:=K[t^2,t^3]$ via the isomorphism induced by the ring homomorphism $K[x,y]\to R$ given by $x\...
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Are the numbers calculated from a log resolution birational invariants?

Let $C = V(x^2 - y^3)$ be the cuspidate cubic which sits inside $\mathbb{C}^2$. Let $\pi: \text{Bl}_0(\mathbb{C}) \to \mathbb{C}^2$ be the blow up of the origin. The reduction of the total transform ...
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2 votes
0 answers
74 views

Irreducible hypersurfaces over $\mathbb{C}$ that are not regular in codimension one

Let $V = V(f) \subseteq \mathbb{C}^n$ be an irreducible (affine) hypersurface. The singular locus $\text{Sing}(V)$ of $V$ is the set of points in $V$ where all the partial derivatives of $f$ vanish. I ...
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1 vote
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Resolution of an ordinary multiple point on an irreducible plane algebraic curve

Let $C \subset \mathbb{A}^2$ be an irreducible plane algebraic curve and $P \in C$ be its ordinary point of multiplicity $m$, i.e., there are exactly $m$ tangents of $C$ at $P$ and they are pairwise ...
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2 votes
0 answers
57 views

Puiseux series and absolutely convergence

Let $f(x,y)\in \mathbf{C}[x,y]$ a plane curve $f(0,0)=0$. It's well knowing that using a Newton's method, we can find a Puiseux series associated with each branch of the curve, i.e., we have an ...
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2 votes
1 answer
51 views

Laurent series and apparent essential singularity

After reading "Mathematics for physicists" by Susan M. Lea I encountered a subtlety that I can't turn my head around (p. 128). Consider function $$f(z)=\frac{1}{z^2-1} = \frac{1}{2}\left[\...
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0 votes
1 answer
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Types and nature of singularities of f(z)=1/z ln(1-z)

Consider the complex function f(z) = (1/z)×ln(1-z) , It seems like having a removable singularity (because,while comparing with corresponding real function,the limit exists.) The function has a branch ...
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4 votes
1 answer
125 views

Regular differentials on a singular curve.

Let $X'$ be an irreducible singular algebraic curve over an algebraically closed field $k$, and let $X \to X'$ be its normalization, and consider a (singular) point $Q \in X'$. Let $K = Q(X)$ be the ...
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0 votes
1 answer
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Does any finite subset of a curve fit into an open affine?

Let $C$ be an irreducible and non-singular algebraic curve over an algebraically closed field $k$, and let $S \subset C$ be any finite set of points. Is it true that there is always an open affine $U \...
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2 votes
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How can I prove that affine hypersurface $V(X^2 + Y^5+Z^5 + 1) \subset \mathbb{A}^3$ is not rational?

$(*)$ I would like to prove that $Spec(\mathbb{C}[X,Y,Z]/\langle X^2+Y^5+Z^5+1 \rangle)$ is not rational (or equivalently that $Proj (\mathbb{C}[X,Y,Z,T]/\langle X^2 T^3+Y^5+Z^5+T^5 \rangle)$ is ...
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2 votes
1 answer
118 views

Show that the image in jacobian has an ordinary double point

I'm solving this problem from 11.12 in Birkenhake C., Lange H. - Complex abelian varieties. Here $W_2$ is an image of $\mu:C^{(2)}\longrightarrow \operatorname{Pic}^2(C)$. Using Rhiemann singularity ...
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1 vote
0 answers
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Trying to understand exceptional divisors and Du Val singularities.

I'v been trying to understand how Du Val singularities are resolved and what the exceptional divisors look like so I can work out their dynkin diagrams. A basic example I tried in the interest of ...
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0 votes
1 answer
151 views

On $K(\pi, 1)$ space.

As far as I know, $K(\pi,1)$ space is a manifold $M$ such that $\pi_n(M) = 0$ for $n > 1$ and $\pi_1(M) = \pi$. Q. Why is the singular cohomology $H_{\mathrm{sing}}^i(M, {\Bbb Z}/n)$ isomorphic to ...
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