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.
364 questions
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Dimension of Jacobian Algebra (in Singularity Theory)
Motivation: In Singularity Theory, the Milnor Number is defined to be the dimension of the Jacobian algebra and is finite in the case of an isolated singularity.
We define the Jacobian algebra of a ...
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Do we have general polynomial equation $f(x,y) = 0$ of degree $d$ which has cusp singular point at origin?
Do we have general polynomial equation $f(x,y) = 0$ of degree $d$ which has cusp singular point at origin?
I want to explicitly compute something in singularity theory for the cusp $X = Z(f) \subset \...
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Two definitions of a normal complex surface
I am looking for the definition of a "normal surface singularity". I could not find such a definition even in the book with title Normal Surface Singularities (https://link.springer.com/book/...
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The link of the zero set of the polynomial $f(z_1,z_2)=z_1^p+z_2^q$ at the origin
Let $p,q\geq 2$ be relatively prime integers and consider the polynomial $f(z_1,z_2)=z_1^p+z_2^q$. For small $\epsilon >0$, let $K$ denote the intersection $f^{-1}(0)\cap S_\epsilon \subset \Bbb C^...
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Contraction of a loop curve on an algebraic surface.
Let $S$ be a smooth algebraic surface over ${\Bbb C}$. Suppose we have a connected curve $C$ whose each irreducible component $C_1, \ldots, C_n$ is ${\Bbb P}^1$, i.e., a rational curve. Suppose that ...
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Methods to show that a singularity in a variety is not a quotient singularity
Let $X$ be a variety (irreducible, normal) over an algebraically closed field $k$ of characteristic $0$, and let $p\in X$ be a singular point. To simplify things, I'll assume that $X$ is affine and ...
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Normal Vectors of Coupled Surfaces
I have two equations:
$$
f(x,y,z_1,z_2) = 0,\\
g(x,y,z_1,z_2) = 0
$$
for two functions of interest $z_1(x,y)$ and $z_2(x,y)$.
Something like:
$$
𝑥\sin(𝑧_1)+𝑦\log(𝑧_1+𝑧_2)=0,\\
𝑥(𝑧_1 𝑧_2)+𝑦\...
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Can a signed distance bound function surface be singular over a set of measure that is non-zero?
Motivation
This is a follow-up from my question here: Can an implicit surface be singular over a set of measure that is non-zero?.
I actually thought that the answer would be positive, but ...
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Can an implicit surface be singular over a set of measure that is non-zero?
Let $f:\mathbb{R}^3\to\mathbb{R}$ be a continuous function such that $S = f^{-1}(0) = \{p\in\mathbb{R}^3\,:\, f(p)=0\}$ is a surface. Let $f$ be continuously differentiable almost everywhere on an ...
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Does local parametrization for pure-dimensional germs require a general position condition?
Background
Following Gunning's book, Introduction to Holomorphic Functions of Several Variables, Vol II, one has the following result (Lemma E.12) given as a corollary of the Local Parametrization ...
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Stability of functions vanishing to 2nd order with respect to perturbations vanishing to 3rd.
The following problem comes from the study of normal crossings degenerations, and a search for an intrinsic, differential geometric definition for normal crossings degenerations in dimension 3.
The ...
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Equivalent definitions of an ordinary cusp
In Multidimensional Real Analysis I, p. 144, it mentioned (without proof) that the two following definitions of an ordinary cusp are equivalent:
Definition 1: If $\phi: I\rightarrow\mathbb{R}^2$ is a $...
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What spatial curves have delta-invariant one?
Is it true that if a germ of a spatial curve $(C,x)\subset(\mathbb{C}^n,x)$ has $\delta=1$ and two irreducible components then it is isomorphic to an $A_1$ singularity?
I take the definition of the ...
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Does maximal contact hypersurface (m.c.h.) exist for any curve over a field of arbitrary characteristic, and if so, for what definition of m.c.h.?
Does a maximal contact hypersurface always exist for any curve over a field of arbitrary characteristic, and if so, for what definition of maximal contact hypersurface?
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Is every normalization a blowup?
Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example.
Let $Y \to X$ be the normalization. The answer is positive in ...
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What condition of the Hawking singularity theorem fails for de Sitter and Minkowski spacetimes?
I am trying to figure out what condition of the Hawking singularity theorem fails for de Sitter and Minkowski spacetimes. Some has to fails otherwise we would had geodesic incompleteness which its ...
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What is the nature of the pole of the derivative of $\frac{1}{1- \ln(x)}$ at $x=0$?
I'm interested in the function $\frac{1}{1-\ln(x)}$ on positive real line.
One can experimentally see that
$$ \lim_{x \rightarrow 0^+} \left[ x \frac{d}{dx} \left[ \frac{1}{1 - \ln(x)} \right] \right] ...
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Restriction of a vector bundle to a nodal curve
Let $S\to B$ be an elliptic surface with one nodal singular fiber C (a nodal projective curve $C$ of genus $g=1$). Let $\mathcal F$ be a slope-semistable rank-$2$ vector bundle on $S$.
What can we say ...
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Reference request non existence of minimal resolution.
In this page of Wikipedia(https://en.wikipedia.org/wiki/Resolution_of_singularities), it writes, the hypersurface in $\mathbb{A}_\mathbb{C}^4$ defined by the equation $xy-zw$ has no minimal resolution....
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Singularity extraction: $\int_{-1}^{1}\int_{-1}^{1}\frac{dxdy}{\sqrt{(x-.3)^2+y^2}}$
$\int_{-1}^{1}\int_{-1}^{1}\frac{dxdy}{\sqrt{(x-.3)^2+y^2}}$ has the following physical meaning:
it is the potential of the uniform surface source distributed on a square $|x|,|y|<1$ observed at a ...
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Calculation regarding divisors and resolution of singularities
I'm trying to understand a calculation on the second page of https://sma.epfl.ch/~filipazz/notes/adjunction_and_inversion_of_adjunction.pdf, and I have a couple questions.
Here is the setup. $X$ is ...
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What are the possible applications in maths and physics of vector fields along smooth maps?
I am currently working on a problem related to singularities of mappings between manifolds with metrics and the interplay of metric singularities with mapping singularities. Given a smooth map $F:M\...
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Does $Spec(k[G/[P,P]])$ have worse than quotient singularities?
Let $k$ be an algebraically closed field of characteristic zero, let $G$ be a connected reductive linear algebraic group over $k$, and let $P$ be a parabolic subgroup of $G$. So we have the flag ...
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Geometric notion of modality
I'm reading Singularity Theory (one of the authors is Arnold). I am a bit confused on the concept of modality. Is there an easy geometric description of it? The book has a relation between codimension,...
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Conditions necessary for the zero set of $Ax^3+Bx^2y+Cxy^2+Dy^3+Ex^2+Fxy+Gy^2+Hx+Iy+J$ to be an irreducible singular curve
This question is about three specific types of irreducible cubic curves in the plane with genus zero. Additionally, any curve which is not the zero set of a polynomial in two variables with real ...
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Transversality of strict transforms
When reading a book, I encountered the following assertion: let $k$ be a field (maybe perfect if it makes things easier), $X$ a smooth $k$-variety and $Y$ a closed irreducible subvariety, also smooth. ...
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Factorial $+$ (at worst) quotient singularities $\implies$ smooth?
Let $X$ be a normal variety (irreducible) over a field $k$ which is algebraically closed and characteristic $0$. Edit: We can add the following assumption, which may help (?): $\mathscr O_X(X)^*=k^*$, ...
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Calculating Milnor number of polynomials
I am trying to understand the Milnor number introduced on Wikipedia.
The second example is for a polynomial $f(x,y) = x^3 +x y^2$ with derivatives $f_{x}=3 x^2 +y^2$ and $f_{y} = 2 x y$. In the ...
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Bypassing a singularity at the zero-frequency in a numerical integral
I am attempting to implement a model outlined in this paper:
General magnetostatic shape–shape interactions
Background
This model allows the calculation of magnetostatic interaction energies between ...
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What is the relationship between irreducibility of a polynomial and integral domain?
On Toni Annala - Resolution of Singularities, I read that "as $y^2 - x^3 - x^2$ is an irreducible polynomial, the coordinate ring $O_C (C)$ is an integral domain, as is the localization $O_{C, p}$...
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Multiplicity of a singular point, Ideals, and Maple/Algorithms
I am teaching myself about algebraic geometry, and the classification of singular points on algebraic curves $f(x,y)=0$, where $x,y\in\mathbb{C}$.
One way to classify these singular points (a set of $...
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Is $0\in \{ux+vy+wz=0\}\subseteq \mathbb C^6$ a quotient singularity?
I am trying to gain some intuition for telling when a variety has quotient singularities.
The example I am focusing on here is the affine variety $X$ which is cut out by the equation $ux+vy+wz=0$ in $\...
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Resolving a Node of a Plane Curve
In the book Algebraic Curves and Riemann Surfaces, Miranda explains how to resolve the node of a plane curve by plugging the hole at the node using hole charts. The idea is at a node $p$, the surface $...
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What Does it Mean for Singularities to be Presented Transversely in the Context of J. Montaldi's PhD Thesis
I was reading the Phd Thesis of J. Montaldi and I came across the following paragraphs:
''In the cases where $k+d<\sigma(k, p)+p$ then we let $\left(W_1, \ldots, W_3\right)$ be the rinite set of $x$...
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Necessary and Sufficient Conditions for Existing an Envelope for a Parametric Family of Implicit Surfaces
Let $F(x, y, z, \lambda)=0$ be a parametric family of implicit surfaces. Sometimes the envelope of the family exists as another surface, but at other times it may degenerate to a curve or a point, or ...
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homology of complex singular curve
Let $C \subset \mathbb{C}^n$ be a singular complex curve. Is there a way to compute its (singular) homology? (or at least its betti numbers / Euler characteristic).
If it were non-singular, then $C$ ...
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Image of a 'narrow' set under a polynomial mapping is a proper semialgebraic subset.
Consider a set $X\epsilon=\{y^2 - \epsilon^2 x^2 \leq 0\} \subset \mathbb{R}^2, 0<\epsilon <<1$
, i.e. a narrow cone passing through the origin. I would like to prove some properties of $f(X\...
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Are varieties normal if and only if they are analytically normal?
Let $V$ be a normal variety, and $p \in V$ a point. It is a theorem of Zariski[1] that the completion $\hat{\mathcal O}_{V,p}$ is a normal ring.
Does the converse also hold? Does $\hat{\mathcal O}_{V,...
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Dominant morphisms between projective varieties
Suppose $f : X\to Y$ is a finite surjective morphism of projective integral complex varieties, and let $g : X'\to X$ and $h : Y'\to Y$ be surjective birational morphisms from smooth projective ...
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Geometric interpretation of the ampleness of the canonical class of a normal algebraic surface
Let $X$ be a minimal, smooth and projective algebraic surface of general type over the complex numbers. Then the ampleness of $K_X$ has a very geometric interpretation: $K_X$ is ample if and only if $...
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On the definition of a simple normal crossing divisor
I would like to ask about the definition of a simple normal crossing divisor. Let me take the definition for instance in Kollar's book Lectures on resolution of singularities.
Let $k$ be a field (one ...
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Single variable critical point degeneracy
I'm reading Singularities of Differential Maps by Arnold, Gusein-Zade, and Varchenko, and I'm a bit confused about their definition of a degenerate critical point. Unlike what I've found on the ...
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Computing the Milnor Number of $x^p+y^q+z^r-xyz$
I would like to compute the Milnor number of $f(x,y,z) = x^p+y^q+z^r-xyz$ which amounts to finding the complex dimension of the underlying vector space of $\mathbb{C}[x,y,z]/(px^{p-1}-yz,qy^{q-1}-xz,...
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How to calculate Delta Invariant of of algebraic curve?
I recently asked a question regarding tangent cones here: Tangent cone of an arbitrary algebraic curve
After doing some reading, I have another question on how to calculate the delta invariant of ...
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Singularitie of type j10
Why the condition $4a^3 + 27\neq0$ in the familie of unimodal singularities $J_{10}$ given by $f_a(x,y)= x^3 + a x^2 y^2 + y^6$. Do I need this condition to prove that this family is 6-$\mathcal{R}$-...
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Tangent cone of an arbitrary algebraic curve
So, my problem is: given a real/complex (I will assume complex) algebraic curve, say $f(x,y)$, or $f(z,w)$ for $x,y\in\mathbb{R}$ or $z,w\in\mathbb{C}$ (I would like to hear your thoughts in either ...
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Discriminant of $R^2 \rightarrow R^2$ map
I want to calculate the discriminant set of a function (germ) $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ based on the $\mathbb{R} \rightarrow \mathbb{R}$ example of this article. The function is ...
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How do I get f' from f that are A-equivalent?
I'm struggling with following problem realted to the singularity theory. I assume that $f, f' \in C^\infty(X,Y)$ and $f$ is a stable mapping. I would like to move from $f$ to some $f'$ which is $A$-...
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Why the maximum number of points of tangency between a line and a generic plane curve is two?
The picture comes from this book Catastrophe theory, page 57.
I can understand it intuitively, but how can we prove it?
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contact between surfaces
Let $s_1(x,y)=(x,y,f_2(x,y))$ and $s_2(x,y)=(x,y,f_2(x,y))$ be two regular surfaces. What is the definition of contact between these 2 surfaces? I read somewhere that the osculating spheres of a ...
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