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# 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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### 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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### 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 ...
1 vote
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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-$\mathcal{R}$-...
1 vote
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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),...
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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 ...
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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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### 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 ...
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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 ...
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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 $r$ ...
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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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### 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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1 vote
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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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