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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36 views

If the vector space dimension of $\mathbb{C}[[x,y]]/I$ over $\mathbb{C}$ is finite, then $I$ contains power of $(X,Y)$

I am trying to understand the proof which goes like this. If $\mathrm{dim}_{\mathbb{C}} \ \mathbb{C}[[x,y]]/I$ is finite, then $\mathbb{C}[[x,y]]/I$ has a finite composition series whose ...
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38 views

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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35 views

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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16 views

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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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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10 views

Examples of a birational map with conditions

I need of an example in the following situation. Let $X$ be an algebraic variety embedded as a closed subvariety of a nonsingular variety $M (i: X \rightarrow M )$. Let $\pi : \tilde{M} \rightarrow M$...
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28 views

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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1answer
40 views

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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48 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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50 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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23 views

sheaf and de Rham cohomology of projective lines glued to order $n$

Let $X = \mathbb{P}^1_k \cup_n \mathbb{P}^1_k$ be the union of two projective lines, glued together at a single point, where the gluing is of order $n$. I would like to compute the sheaf cohomology ...
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30 views

Calderon-Zygmund in $L^p$ with $2<p<+\infty$

Let $T$ be the following function: \begin{equation*} Tf(x)=P.V\int_{\mathbb{R}^n}\frac{\Omega(y)}{|y|^n}f(x-y)\,dy=\lim_{\varepsilon \rightarrow 0^{+}} \int_{|y|>\varepsilon} \frac{\Omega(y)}{|y|^n}...
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42 views

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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42 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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20 views

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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28 views

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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73 views

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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34 views

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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40 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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Adjacencies of simple singularities

I am currently studying the concept of adjacency in singularity theory. We say that singularity $f$ is adjacent to $g$ if there exists a one-parametric family of functions $F_t(x,y), t \in [0,1]$ such ...
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29 views

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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29 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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30 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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1answer
79 views

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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1answer
88 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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38 views

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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90 views

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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10 views

How do you find this parametrization of (2,3) knot?

I have started reading Milnor's 'Singular points of complex hypersurfaces' and in page 3 and 4, he states the following example. Let $f(z,w) = z^p + w^q$ where $p,q$ are coprime. It has a singularity ...
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109 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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55 views

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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1answer
71 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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72 views

Milnor number/Mapping Degree

I am reading in John Milnor's Book Singular Points of Complex Hypersurfaces and struggle to do a similar computation he did. Namely, how can I compute explicitly the Milnor number $\mu(f)$ on the ...
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39 views

Does any function germ belong to its gradient ideal?

Consider the $\mathbb C$-algebra $\mathcal O_2$ of holomorphic function germs on $\mathbb C^2$ at the origin $0$. Is it true that any germ $f\in\mathcal O_2$ with $f(0)=0$ belongs to $(\partial f/\...
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67 views

Polynomials with nondegenerate critical points and their derivatives

I have the following problem (it arises in a calculation in Topological String Theory in the paper here that the author glosses over when evaluating a spectral sequence, but condenses down to this ...
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40 views

Smooth maps to singular sets

Let $X\subseteq\mathbb R^n$ be an affine variety (that is, a set given by polynomial equations). One may suppose $X$ singular (the case of non-singular $X$ is obvious). Is it true that any continuous ...
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1answer
181 views

Contour integral around an essential singularity

The complex function $$f(z)=\frac{1}{e^{1/z}-1}$$ has an essential singularity at $z=0$, and an infinite quantity of poles inside every open neighborhood containing it. Let $\mathbb{R}\ni\epsilon>...
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33 views

Is the desingularization of this surfaces of general type?

I have a family of deformation $\mathcal{X}\to B$ of a surface $S$. The surface has a finite number of singular point and it is normal. There is a divisor $D\subset B$ such that the surface $\mathcal{...
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1answer
29 views

Prove that for for an arbitrary linear operator all eigenvalues lie in a circular ring.

Prove that for for an arbitrary linear operator all eigenvalues lie in a circular ring. {z $\in$ $С$ | $\sigma_n(A)$ $\leq$ $|z|$ $\leq \sigma_1(A)$}
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1answer
124 views

Unknot: embedded vs immersed bounding disk

Suppose that we have a knot $K\subset \mathbb{S}^3$. If $K$ bounds* an embedded 2-disk then $K$ is the unknot. But what happens if $K$ bounds an immersed 2-disk? The immersed disk generically will ...
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123 views

the range of the entire function $ e^z(1+\cos\sqrt{z} ) $ Picard's Great theorem for

Let $$f(z) = e^z (1+\cos\sqrt{z} ) $$ $\Omega=\{z\in\Bbb C: |z|\gt r\}$, $r\gt 0$. What is $f(\Omega)$? where $\sqrt{z}=\exp{(\text{Log }z/2)}, \text{Log }z=\log|z|+i\arg z,\arg z\in(-\pi,\...
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1answer
154 views

Tangent cone of a non-isolated singular point

Assume $X\subset \mathbb P^n$ is a variety (Edit: let's say $X$ is a hypersurface in $\mathbb P^n$, as pointed out in the comment) and $x\in X$ is a singular point which is not isolated. Intuitively, ...
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1answer
71 views

Isolated critical points and the Milnor number

I was looking at the Wikipedia page of the Milnor number found here, and specifically tried to work through proving for myself one of its stated properties. It mentions the Milnor number is finite if ...
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1answer
158 views

Exceptional divisor of the blow-up of affine cone at the vertex

Let $f(x) \in \mathbb{C}[X_1,...,X_n]$ be a homogeneous polynomial in $n$ variables such that the zero locus $V$ of $f$ in $\mathbb{C}^n$ is singular only at the origin. Denote by $\pi:\widetilde{V} \...
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1answer
45 views

Reference request: Knots that don't come from Milnor spheres.

In Milnor's book "Hypersurface singularities" He discusses shortly knots that arrive as Milnor spheres of algebraic curves, i.e knots that are the intersection of a $3$ sphere around a singular point ...
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41 views

Show that if $f \sim^{\mathcal{R}} g$, then $\text{ord } (f) = \text{ord } (g)$.

Problem: Let $f,g \in \mathbb{C}[[\mathrm{x}]] = \mathbb{C}[[x_1,\dots,x_n]]$. Show that if $f \sim^{\mathcal{R}} g$ then $\text{ord } (f) = \text{ord } (g)$. My attempt: Follow the definition, we ...
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37 views

Complex structure deformations of toric Calabi-Yau 3-folds

If $X$ is a toric Calabi-Yau 3-fold (so, it is in particular, non-compact), is it necessarily true that $h^{2,1}(X)$ vanishes? (Such a CY 3-fold is termed ``rigid''.)
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209 views

Can we approximate a vector field on the plane with non-vanishing vector fields in $L^2$?

Let $V$ be a compactly-supported smooth vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated. Does there exist a sequence ...
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1answer
41 views

What is the name of this type of object?

I am interested in the following type of objects (I will describe them later), but because I am very new to the subject of differential geometry, and I just have a very basic notion of the concepts, I ...
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61 views

Calculating infinitely near points of a curve singularity

Let $f(x,y)$ be a polynomial in $\mathbb{Q}[x,y]$ with a singular point at the origin $(0,0)$. Currently I am writing a Maple program to compute its iterated blow-up embedded resolution as a hierarchy ...
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2answers
188 views

Deciding whether a point on a surface is singular

Consider $S^2 \in \mathbb{R}^2$. It is well known that it is a regular surface. We can parametrize (a patch of it) by (EDIT: this seems wrong, see comment) $$x(u,v) = \cos u \cos v$$ $$y(u,v) = \sin ...

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