The singularity-theory tag has no wiki summary.

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### When is a singular point of a variety smooth?

If $X$ is a nonsingular algebraic (or analytic) variety over $\mathbb C$ or $\mathbb R$ then it is certainly $C^\infty$ over the reals.
The converse is false for a silly reason : in the real or ...

**27**

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**3**answers

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### Wanted: example of a non-algebraic singularity

Given a finitely generated $\def\CC{\mathbb C}\CC$-algebra $R$ and a $\CC$-point (maximal ideal) $p\in Spec(R)$, I define the singularity type of $p\in Spec(R)$ to be the isomorphism class of the ...

**26**

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### Microlocal geometry - A theorem of Verdier

(1) In "Geometrie Microlocale", Verdier states the following theorem.
Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$.
Then for $\ell$ a linear form on $E$, we have a ...

**26**

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**2**answers

1k views

### Limit of a series of singularities

The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities ...

**23**

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**7**answers

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### Examples of Mixed Hodge Structures

Does anyone know a user-friendly, example-laden introduction to mixed Hodge structures? I get from Wikipedia how to calculate for a punctured and pinched curve ...

**18**

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**5**answers

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### Is a 'generic' variety nonsingular? Or singular?

I'd like to know whether there's some coherent meaning of 'generic' for which one can say that a 'generic' variety over an algebraically closed field $K$, say, is nonsingular or singular. We could ...

**13**

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**2**answers

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### Elementary short exact sequence of sheaves

This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow ...

**13**

votes

**1**answer

347 views

### Higher Cerf Theory

Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that ...

**12**

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**4**answers

979 views

### What formal properties should resolution of singularities have?

If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...

**12**

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675 views

### Comparing fundamental groups of a complex orbifolds and their resolutions.

Let $X$ be a complex manifold with quotient singularities, and let $\tilde X$ be its resolution (that exists, for example, by Hironaka). Then I am pretty sure that $\pi_1(X)\cong \pi_1(\tilde X)$.
...

**11**

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**4**answers

988 views

### Singular semi-Riemannian Geometry: usefulness and state of the art

My question has two parts, one concerning the state of the art of the subject, and the other the usefulness.
1. State of the art.
Can someone provide references reflecting the state of the art in ...

**11**

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**1**answer

412 views

### Can a PDE constrain the degree of a $C^\infty$ map germ?

Let $\pi:E\to M$ be a smooth vector bundle over a smooth manifold, with $\text{rank}(E)=\text{dim}(M)$. For a section $\sigma$ of $E$ with a zero at $p\in M$, define the degree of the zero at $p$ to ...

**10**

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**2**answers

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### Giant Rat of Sumatra singularity

I would be grateful for explanations of the issues raised in any
of these three questions, or pointers to the relevant literature
(now updated with answers):
How did a particular singularity come ...

**10**

votes

**1**answer

250 views

### Analogue of singularity theory in other categories

Whitney, Thom, Mather, Arnold and others develoved the singularity theory of smooth maps.
Does there exist any analogue of this theory in the category of TOP or PL (or Lipschitz) maps?
I mean notions ...

**9**

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**1**answer

240 views

### Do level sets always correspond to even graphs?

Suppose I have a level set of some function $f\colon\mathbb{R}^n\rightarrow\mathbb{R}^m$, say $L:=\{x:f(x)=c\}$. Let $S$ denote the points in $L$ at which $L$ is locally diffeomorphic to an open ...

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**3**answers

520 views

### Singular fibers of generic smooth maps of negative codimension

This is in some sense a follow-up to my question on submersions.
Let $f\colon\thinspace M\to N$ be a generic smooth map between closed manifolds of dimensions $m$ and $n$. Assume that the codimension ...

**9**

votes

**1**answer

955 views

### Du Val singularity of type G=A,D,E and “small” representations of G

We all know that a simple singularity $W_G(x_1,x_2,x_3)=0$ of type G=A,D,E has the following nice deformation involving the Cartan subalgebra $\mathfrak{h}$ of the Lie algebra $\mathfrak{g}$ of $G$. ...

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232 views

### Can I compute the cohomology of the complement of a log canonical divisor as if it were normal crossings?

Let $X$ be a smooth projective variety and $D$ a log-canonical divisor and let $U = X \setminus D$. I have heard the slogan "log-canonical is just as good as normal crossings for Hodge theory". This ...

**8**

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**2**answers

626 views

### Whitney stratifications

Many results on characteristic classes of singular varieties (as well as other singularity-theoretic constructions) make use of a so-called "Whitney stratification" of the variety under consideration, ...

**8**

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**3**answers

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### Singularities of pairs

In the next days I have to give a talk in which I need to explain some of the usual singularities of pairs that one meets when dealing with the minimal model program: KLT, DLT and LC pairs.
In ...

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**2**answers

487 views

### Small neighborhoods of singularities on varieties

In Singular points of complex hypersurfaces, John Milnor proves the following theorem:
Let $x \in V$ be a point on a variety $V$ in $\mathbb{R}^n$ or $\mathbb{C}^n$. Assume $x$ is either a smooth ...

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**1**answer

373 views

### Why can you deform singularities in two dimensions but not in higher dimensions?

I've been trying to read this paper to understand deformations of surface quotient singularities. I'm particularly interested in when one can deform certain cyclic quotient singularities into other ...

**8**

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**3**answers

394 views

### Applications for knowing the singularities parametrized by the boundary of a moduli space

Given a moduli space $M$ of some smooth algebraic geometric object such as curves, surfaces, etc. Let $\overline{M}$ be a compactification of $M$. Then, $\overline{M}\setminus M$ introduces ...

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272 views

### Cohomology and conifold transition for the quintic

Let $Y\subset \mathbb{C}P^4$ be the quintic threefold given by the equation $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4+5X_0X_1X_2X_3X_4=0$$
it has 125 singular points whose links are homeomorphic to $S^2\times ...

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291 views

### Which presentations of (non)planar algebras give rise to knots?

Reidermeister's theorem states that the set of knots, modulo ambient isotopy, is isomorphic to the planar algebra generated by crossings, modulo Reidemeister moves. This planar algebra presentation is ...

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**1**answer

324 views

### When two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$ are the same?

Let us consider two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$, where $G$ and $G'$ are finite subgroups of $\mathrm{GL}(n,\mathbb{C})$ acting linearly.
It is easy too see, that a different ...

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639 views

### Implicit function theorem at a singular point?

Let $F:\mathbb{R}^2 \rightarrow \mathbb{R}$ be three times continuously differentiable in some open neighborhood $\mathcal{U}$ of $(0,0)$. Suppose that $F(0,0) = F_x(0,0) = F_y(0,0) = F_{xy}(0,0) = 0$ ...

**7**

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410 views

### Resolution of singularities for flat families.

Is there a resolution of singularities for flat families?
More precisely, if $X \rightarrow \mathbb{A} ^n$ is a flat map, does there exist a map $Y \rightarrow X$ such that, for every $p \in ...

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418 views

### Factoriality: local or global?

Let $X$ be an algebraic variety. I have read the following definitions:
$X$ is factorial if every Weil divisor on $X$ is Cartier.
$X$ is locally factorial if all its local rings are unique ...

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**1**answer

668 views

### Rational singularities for fibered surfaces

This question consists of two parts. I will try to be as short and clear as possible.
Let $S$ be a Dedekind scheme of characteristic zero. The main examples are $\mathbf{P}^1_k$, with $k$ a field of ...

**7**

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**1**answer

290 views

### Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...

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**1**answer

422 views

### Simplified treatment of resolutions of complex analytic varieties?

According to the article of Hauser:
The Hironaka theorem on resolution of singularities http://www.ams.org/journals/bull/2003-40-03/S0273-0979-03-00982-0/home.html
The existence of resolution of ...

**7**

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286 views

### Singularities arising from the Minimal Model Program (an algebraic point of view)

I will start the story by the end:
Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ...

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441 views

### Physicists Euler number conjecture

Physicist's Euler number conjecture says:
If $G \subset SL(n,\mathbb{C})$ is a finite group, $X=\mathbb{C}^n/G$ is the quotient space and $f:Y \rightarrow X$ a crepant resolution (always exists for ...

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867 views

### Is there an obvious way for showing singularities are quotient?

I'm stuck on a technicality concerning singularities.
Basically, I have to show that the singularities of a $\mathbf{certain}$ normal projective variety over $\mathbf{C}$ are rational. (I won't ...

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**2**answers

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### construct the elliptic fibration of elliptic k3 surface

Hi all,
As we know, every elliptic k3 surface admits an elliptic fibration over $P^1$, but generally how do we construct this fibration? For example, how to get such a fibration for Fermat quartic?
...

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1k views

### Resolution of singularities

What is the relation between crepant resolutions and minimal resolutions? Are they the same thing?

**6**

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765 views

### “Arithmetic genus” of a plane curve singularity.

I believe that the following questions are very basic, but I don't know how to get a reference.
Consider a curve in the plane $C\in \mathbb C^2$ with a singularity at $0$ and suppose it is
...

**6**

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**4**answers

803 views

### I was wondering if the set of singular loops is a (somewhere) submanifold of loop space?

The set of all smooth maps $S^1\to M^n$ ($M$ is a smooth manifold) is a generalized manifold(see http://ncatlab.org/nlab/show/smooth+loop+space).
I was wondering if the set of singular loops (maps ...

**6**

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**3**answers

607 views

### hyperplane sections of isolated hypersurface singularities.

Given an isolated singularity $p$ in a hypersurface $Y$ of dimension $n$ (let say a surface in $\mathbb{P}^3$). I can intersect $Y$ with a hyperplane $H$ passing through $p$ such that it induces a ...

**6**

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**1**answer

517 views

### Bertini's Theorem

Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...

**6**

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**1**answer

100 views

### Properties of singularities that are preserved by categorical quotients

Let $G$ be a reductive group acting on an affine singular
variety $X$, and let $X/G$ be the categorical quotient. I know that if
$X$ has rational singularities, then so does $X/G$ ...

**6**

votes

**1**answer

148 views

### Is there an algorithm to find out the number of small solutions to a polynomial equation, when we vary all the coefficients?

Let $\Phi (z,t)$ be a polynomial given by
$$ \Phi(z,t) := z^n + A_{n-1}(t) z^{n-1} + \ldots + A_1(t) z + A_0(t).$$
Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation
$$ ...

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437 views

### What strict resolutions of singularities are needed?

Suppose we have a collection, $S$, of singularities types and consider a resolution of singularities (this is: a proper birrational morphism $Y\rightarrow X$ such that Y only contains singularities of ...

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268 views

### Whitney stratification and affine grassmanian

Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by ...

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### What does being Analytically Isomorphic imply for classification of singularities on curves?

Hartshorne I.5 mentions the definition of being analytically isomorphic:
P on X and Q on Y are analytically isomorphic iff the completion of O_P is isomorphic to the completion of O_Q where the ...

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491 views

### Blowing-up an ordinary double point, then contracting the exceptional locus to a curve

Let $X\subset\mathbb P^4$ a projective hypersurface with an ordinary double point at $o\in X$.
Blow-up $\mathbb P^4$ at $o$ and let $E\simeq\mathbb P^3$ the exceptional divisor of this blow-up. ...

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558 views

### Is the desingularization of a normal variety with only quotient singularities projective

The base field will be the field of complex numbers. I have a slightly technical problem concerning the resolution of singularities of a certain variety. Basically, I want to to know if it is ...

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398 views

### How can we find a surface with a given singularity?

I was surprised the first time I learned that a quintic plane curve can have an $A_{10}$ singularity i.e $x^2+y^{10}$. I am wondering if there is something about that phenomenon: Given a singularity ...

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**1**answer

693 views

### Localization of vanishing cycles

Consider a regular holonomic D-module (or a perverse sheaf) $M$ on a smooth variety $X$. Let $f:X\to A^1$ be a polynomial (or holomorphic) function.
Question: Is it true that the $\lambda \in A^1$ ...