Skip to main content

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.

Filter by
Sorted by
Tagged with
3 votes
0 answers
74 views

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 ...
SeparatedScheme's user avatar
0 votes
0 answers
35 views

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 ...
darkside's user avatar
  • 589
0 votes
1 answer
55 views

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] ...
Sidharth Ghoshal's user avatar
0 votes
0 answers
25 views

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 ...
Conjecture's user avatar
  • 3,210
4 votes
1 answer
61 views

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....
George's user avatar
  • 289
0 votes
0 answers
85 views

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 ...
Aria's user avatar
  • 422
0 votes
1 answer
36 views

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 ...
EJAS's user avatar
  • 185
0 votes
1 answer
129 views

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\...
Siddharth Panigrahi's user avatar
0 votes
0 answers
83 views

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 ...
Dave's user avatar
  • 13.6k
2 votes
0 answers
43 views

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,...
quantum's user avatar
  • 1,667
0 votes
0 answers
21 views

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 ...
Simon M's user avatar
  • 727
1 vote
0 answers
44 views

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. ...
Alexey Do's user avatar
  • 2,129
0 votes
1 answer
88 views

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 (?): $\mathscr O_X(X)^*=k^*$, ...
Dave's user avatar
  • 13.6k
1 vote
0 answers
55 views

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 ...
Shasa's user avatar
  • 111
4 votes
0 answers
55 views

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 ...
JasonC's user avatar
  • 81
0 votes
1 answer
60 views

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 $O_{C, p}$...
yuuu's user avatar
  • 165
2 votes
1 answer
96 views

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 $...
Sora8DTL's user avatar
  • 107
1 vote
1 answer
129 views

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 $\...
Dave's user avatar
  • 13.6k
0 votes
0 answers
64 views

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 $...
The Special One's user avatar
2 votes
0 answers
36 views

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 $x$...
Sarah Chofakin 's user avatar
0 votes
0 answers
44 views

Type of singularity at infinity for Faddeeva function, error function?

I am a bit confused with the type of singularity at infinity for the following function. $f(z) = z^2e^{(z-1/2)^2}\text{erfc}(z-1/2)$. Alternatively, we can also use the Faddeeva function to re-write ...
Ranger's user avatar
  • 63
0 votes
0 answers
51 views

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 ...
Bumblebee's user avatar
  • 18.3k
0 votes
1 answer
77 views

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 $C$ ...
Serge the Toaster's user avatar
1 vote
0 answers
58 views

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\...
John Doe's user avatar
1 vote
1 answer
73 views

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,...
red_trumpet's user avatar
  • 8,951
2 votes
2 answers
127 views

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 ...
user avatar
1 vote
0 answers
58 views

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 $...
Srinivasa Granujan's user avatar
1 vote
1 answer
520 views

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 ...
Alexey Do's user avatar
  • 2,129
0 votes
1 answer
28 views

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 ...
Redcrazyguy's user avatar
1 vote
0 answers
58 views

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,...
inkievoyd's user avatar
  • 1,825
1 vote
0 answers
244 views

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 ...
Sora8DTL's user avatar
  • 107
1 vote
0 answers
33 views

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-$\mathcal{R}$-...
0212user's user avatar
  • 379
1 vote
1 answer
189 views

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 ...
Sora8DTL's user avatar
  • 107
0 votes
1 answer
118 views

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 ...
zltn.guba's user avatar
  • 123
0 votes
0 answers
51 views

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 $A$-...
Bob's user avatar
  • 1
0 votes
0 answers
84 views

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?
Andy's user avatar
  • 101
0 votes
0 answers
62 views

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 ...
User0212's user avatar
2 votes
1 answer
112 views

How to show $x_0^2+x_1^2+x_2^2=0 \subset \mathbb{CP}^2 \iff \mathbb{CP}^1$

I am currently trying to blow-up an $A_n$ singularity defined by the hypersurface equation: \begin{equation} z_1^2+z_2^2+z_3^{n+1}=0 \subset \mathbb{C}^3 \end{equation} Let $x_i, i=0,1,2$ denote the ...
Anton B's user avatar
  • 23
1 vote
1 answer
130 views

Reasoning behind Fields with "holes"

For this vector field $$F(x, y) = (-\frac{y}{(x+1)^2+y^2} - \frac{y}{(x-1)^2+y^2}, \frac{x+1}{(x+1)^2+y^2} + \frac{x-1}{(x-1)^2+y^2})$$ I'm asked to check if it is a gradient field in region $D = \{(x,...
ludicrous's user avatar
  • 653
3 votes
0 answers
63 views

Locally isomorphic singularities = Locally isomorphic minimal blow-ups?

Let's say that one has two varieties $V_1, V_2$, with singularities at points $x_1$ and $x_2$ respectively, and that there exist open neighbourhoods $U_1$ and $U_2$ of these singularities such that $...
StormyTeacup's user avatar
  • 1,992
0 votes
2 answers
66 views

Singularities of $f(z)=z /\ (\cos(z)-1)$

I'm having difficulties with this function's singularities. As far as I understand $f(z)$ has order $2$ poles in $z=2\pi k$ (where $k$ is integer), but I'm not sure about $z=0$ since it turns ...
fehawq's user avatar
  • 1
4 votes
0 answers
112 views

Number of lines on a singular cubic surface

A smooth cubic surface contains 27 lines, but a singular cubic surface with rational double points contains fewer lines. Question: Why is the number of lines equal to $\binom{8-r}{2}+n-1$, where $r$ ...
LeechLattice's user avatar
5 votes
1 answer
509 views

Definitions of Milnor number

Let $f:\mathbb{C}^n\to\mathbb{C}$ be a function with an isolated singularity at $0$, by which I mean $f'(0)=0$ and there is some $\epsilon > 0$ such that $f'(p)\not = 0$ for all $p$ nonzero with $|...
UniversalConfusion's user avatar
2 votes
1 answer
130 views

Show $y^4 - x^5$ is irreducible in $\mathbb C[[x,y]]$ [duplicate]

Is there a 'conceptual' way to see that $f(x,y) = y^4 - x^5$ is not the product of two power series $a(x,y)$ and $b(x,y)$ unless either $a$ or $b$ are invertible? I guess I am thinking of $\mathbb{C}[[...
UniversalConfusion's user avatar
1 vote
1 answer
121 views

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 ...
Matthias's user avatar
  • 797
0 votes
1 answer
334 views

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 ...
Jo Wehler's user avatar
  • 2,293
2 votes
0 answers
75 views

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 \...
user avatar
4 votes
1 answer
103 views

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 ...
user56980's user avatar
  • 259
0 votes
1 answer
44 views

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 ...
prolea's user avatar
  • 2,032
1 vote
1 answer
369 views

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 ...
prolea's user avatar
  • 2,032

1
2 3 4 5
8