The singularity-theory tag has no usage guidance.

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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 ...

**28**

votes

**0**answers

2k views

### 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 ...

**27**

votes

**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**

votes

**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 ...

**25**

votes

**7**answers

3k views

### 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**

votes

**5**answers

2k views

### 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**

votes

**2**answers

3k views

### 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

**5**answers

743 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 $\pi_1(X)\cong \pi_1(\tilde X)$.
...

**13**

votes

**1**answer

439 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**

votes

**4**answers

1k 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 ...

**11**

votes

**4**answers

1k 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**

votes

**1**answer

446 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**

votes

**2**answers

508 views

### Analytical formula for topological degree

At the first page of the following article http://arxiv.org/pdf/1004.1018v1.pdf [edit: the formula on the arXiv differs from the formula in the published paper, and the formula displayed below is the ...

**10**

votes

**2**answers

1k views

### 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

264 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 ...

**10**

votes

**0**answers

244 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 ...

**9**

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**2**answers

2k views

### 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?
...

**9**

votes

**2**answers

333 views

### Which weighted projective spaces (and their finite quotients) are local complete intersections?

Let $G$ be a finite subgroup of $\textrm{Gl}_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of ...

**9**

votes

**3**answers

583 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

336 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 ...

**9**

votes

**1**answer

968 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 $G$. ...

**8**

votes

**2**answers

1k 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
...

**8**

votes

**2**answers

730 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 $F(0,0) = F_x(0,0) = F_y(0,0) = F_{xy}(0,0) = 0$ ...

**8**

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**2**answers

894 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**

votes

**1**answer

425 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**

votes

**1**answer

250 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 ...

**8**

votes

**1**answer

572 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 ...

**8**

votes

**3**answers

1k views

### 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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505 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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329 views

### Implicit Function Theorem on Singular Varieties

Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function
$$f\colon X \to Y $$
and assume that it is bijective at the level of ...

**8**

votes

**3**answers

435 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 ...

**8**

votes

**1**answer

733 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 ...

**8**

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340 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 ...

**8**

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**0**answers

299 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 ...

**7**

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**1**answer

335 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 ...

**7**

votes

**2**answers

468 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 ...

**7**

votes

**1**answer

545 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 $p_1,...,p_n$. ...

**7**

votes

**1**answer

439 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**

votes

**0**answers

162 views

### Singularities of an analytic function over a non-archimedean field

What do we know about the types of singularities that a convergent power series over a non-archimedean field can have?
More specifically:
i) What types of essential singularities can occur?
ii) Are ...

**7**

votes

**0**answers

340 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 ...

**7**

votes

**0**answers

452 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 ...

**6**

votes

**4**answers

1k views

### minimal resolution of singularities

What is the minimal resolution of singularities of the surface
$S^2(X^3+Y^3+Z^3)-3(S^2+T^2)XYZ=0$ which is a subset of $\mathbb{P}^1\times\mathbb{P}^2$
Please note that in this equation ...

**6**

votes

**3**answers

907 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 ...

**6**

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**2**answers

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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**4**answers

822 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**

votes

**1**answer

327 views

### The singularity of the algebraic stack and the singularity of the coarse moduli space

It is possible that an algebraic stack is smooth while the coarse moduli space is not smooth. I want to know what is relationship between the singularity of the algebraic stack and that of its coarse ...

**6**

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649 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**

votes

**1**answer

109 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

149 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
$$ ...

**6**

votes

**3**answers

446 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 ...