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.
330
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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\...
- 11
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1
answer
41
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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,...
- 6,729
2
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2
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53
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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 ...
- 165
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37
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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 $...
- 1,240
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26
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Deformation to normal cone of the exception divisor of a log-resolution
I have posted this question on MO but I think it is more suitable to post it on MSE. Thanks in advance for any help.
I am reading the paper Iterated vanishing cycles, convolution, and a motivic ...
- 1,226
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1
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147
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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 ...
- 1,226
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1
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22
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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 ...
- 295
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41
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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,...
- 1,675
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84
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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 ...
- 65
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28
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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-$\mathcal{R}$-...
- 369
1
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1
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86
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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 $\in\mathbb{C}$ (I would like to hear your thoughts in either case),...
- 65
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52
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Continuity of the volume of semi-algebraic sets
Let $n$ be a positive integer, and $I$ an open bounded (small) interval of $\mathbb{R}$ parametrized by a variable $y$. We fix a finite number of real polynomials in $n$ variables with coefficients ...
- 1
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1
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104
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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 ...
- 123
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45
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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 $A$-...
- 1
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82
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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?
- 101
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43
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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 ...
- 41
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1
answer
85
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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 ...
- 23
1
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1
answer
81
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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,...
- 633
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0
answers
56
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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 $...
- 1,798
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2
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52
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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 ...
- 1
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89
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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 $r$ ...
- 326
5
votes
1
answer
198
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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 $|...
1
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1
answer
108
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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}[[...
1
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1
answer
74
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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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146
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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 ...
- 2,205
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69
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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 $d-1$:
$$\mathrm{rank} (dF_{p}) = d-1 \quad \text{for all $p \...
- 1,940
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49
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The commutator map
I am trying to study the commutator map of a given Lie group $G$:
$$\mu : G\times G\to G,\ \mu(x,y)=[x,y]=xyx^{-1}y^{-1}$$
I am interested in:
Its singular points (where the the differential is not ...
- 307
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1
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43
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From differential form to vector fields
In page 13 of this course https://www.sissa.it/fa/download/publications/remizov.pdf
from the equation
$$
(F_x+pF_y) dx + F_pdp=0 \qquad (*)
$$
I do not understand how we can get
$$
\dot x =F_p, \quad
...
- 1,942
1
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1
answer
202
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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 ...
- 1,942
2
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1
answer
93
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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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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{...
- 13.6k
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35
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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 ...
- 417
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29
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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},...
- 456
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31
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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 $A_{1}$ ...
- 384
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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,...
- 456
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1
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114
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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 ...
2
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0
answers
74
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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 ...
- 1,854
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0
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53
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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 ...
- 73
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0
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35
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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 ...
- 823
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69
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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 ...
- 1,269
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2
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86
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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 ...
- 685
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1
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368
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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 ...
- 346
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0
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212
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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 ...
- 1,675
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0
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63
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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\...
- 456
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votes
0
answers
73
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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}\...
- 456
0
votes
1
answer
81
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 $I$, ...
- 425
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0
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68
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 = \...
- 456
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0
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36
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 ...
- 1,169
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0
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78
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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 ...
1
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0
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51
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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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