# Questions tagged [singularity-theory]

Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.

482
questions

4
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0
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84
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### Blowing-up a non reduced fiber

Let $X\rightarrow \mathbb{P}^2$ be a smooth conic bundle with a non reduced fiber $F$, and $\widetilde{X}$ the blow-up of $X$ along $F$ with exceptional divisor $F\times\mathbb{P}^1$.
I expect $\...

2
votes

0
answers

66
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### Jacobian ideal as primary idea;

Let $R = \mathbb{C}\{x_1, \dots,x_n\}$ be the ring of germs of analytic functions and let $f \in R$ be a homogeneus polynomial of degree $p$ such that $\sqrt{Jac(f)}=\langle x_1, \dots, x_{n-1}\rangle ...

1
vote

1
answer

83
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### Cohen-Macaulyness of Milnor algebra

Denote by $R = \mathbb{C}\{x_1, \dots, x_n\}$ the ring of germs of analytics maps at the origin in $n$ variables and let $f \in R$ such that $Sing(V(f))=V(x_1, \dots, x_{n-1})$ as sets. In addition, ...

2
votes

1
answer

93
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### Surfaces with rational double points

Let $S\rightarrow \mathbb{P}^1$ a surface fibered in conics over a field. Assume that $S$ has a single non reduced fiber $F$ with two points of type $A_1$ on it.
Blowing-up the two points and ...

4
votes

1
answer

126
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### Singularities of surfaces fibered in rational curves

Let $S$ be a projective surface with a morphism $S\rightarrow\mathbb{P}^1$ whose fibers are either smooth $\mathbb{P}^1$'s or the union of two smooth $\mathbb{P}^1$'s intersecting in a point.
...

2
votes

0
answers

85
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### Is a function looking like a cubic cusp globally equivalent to the cubic cusp?

Let's consider a family of smooth odd functions $\phi_v(u)\colon \mathbb{R}^2\to\mathbb{R}$, which ,looks like' a family of functions $f_y(x)=x^3-yx$ in the vicinity of $(0,0)$: $\phi_v(u)$ has no ...

1
vote

0
answers

127
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### Homogeneous deformation of isolated singularities

Let $f\in \mathbb{C}[x_1, \dots,x_n]$ be a homogeneous polynomial of degree $p$ and let $F \in \mathbb{C}[x_1, \dots,x_n,t]$ be a polynomial such that $F(x_1, \dots, x_n,0)=f$, for every $t_0$ we have ...

2
votes

1
answer

165
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### Deformation of isolated singularities and non zero divisors

Consider $f \in \mathbb{C}\{x_1,\dots,x_n\}$ such that $(V(f),0)$ has an isolated singularity.
Let $F \in \mathbb{C}\{x_1,\dots,x_n,t\}$ be a deformation of $f$ such that there exists some integer $m$...

2
votes

0
answers

53
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### Maximal number of certain types of isolated singularities on $3$-folds

Let $X$ be an algebraic surface with only canonical singularities and such that $K_X$ is nef. Then Miyaoka's "The Maximal Number of Quotient Singularities on Surfaces with Given Numerical ...

-1
votes

1
answer

82
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### Singularities of Painlevé II

It is known that Painlevé II has only simple poles $t_n$ as singularities and these poles have non-movable property, i.e., they depend only on the equation rather than initial conditions. One can try ...

2
votes

0
answers

90
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### Confusion with terminology: Crepant resolution of terminal singularities

In Theorem 1.1 of this article, Bridgeland proves derived equivalence between Crepant resolution of threefold terminal singularities. I am a little confused with this terminology. In particular, a $\...

4
votes

0
answers

135
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### Vanishing cycles and injectivity of the specialisation map

Consider a proper algebraic map between complex varieties $f : X \to D$ ($D$ is the unit disk), which is a submersion over $D^*$. I would like to know if they are any condition on $f$ such that the ...

4
votes

1
answer

163
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### Do there exist linear relations between exceptional divisors

Let $X$ be an isolated, Gorenstein singularity of dimension at least $2$ and $\pi: \widetilde{X} \to X$ be a resolution of singularities. Let $E$ be the exceptional divisor and $E_1,...,E_r$ be the ...

1
vote

0
answers

56
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### Prove or disprove the positivity of the ess inf of a singular function

Consider a measurable radial function $u:\Bbb R^d\to(0,\infty)$ such that
$$\int_{B_\delta(0)} u(x) d x=\infty\quad\forall\,\, \delta>0.$$
I would like to prove or to disprove that there exists $r&...

2
votes

0
answers

80
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### Solutions of the vector field $D=A\frac{\partial}{\partial X}+B\frac{\partial}{\partial Y}$ with $A,B\in k[[X,Y]]$

I am trying to understand the article Reduction of Singularities of the Differential Equation $Ady=Bdx$ by Arno van den Essen.
Let $A,B\in k[[X,Y]]$ be formal power series and $D$ the vector field $A\...

3
votes

1
answer

94
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### du Val singularities in Magma

Is there any way to decide whether a singularity of a surface embedded in $\mathbb{P}^5(\mathbb{Q})$ is a du Val/rational double point in Magma?
Any help is much appreciated.

3
votes

0
answers

139
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### Stability of nodal hypersurfaces

We denote by $\Pi_{n,d}$ the space of homogeneous polynomials of degree $d$ in $n+1$ variables $x_0,\ldots,x_n$, i.e. $\Pi_{n,d}=\Gamma(\mathbb{P}^n(\mathbb{C}),\mathcal{O}(d))$. The group $G=SL(n+1)$ ...

4
votes

2
answers

437
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### Smoothness of fibers over finite fields

Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties over a finite field of characteristic different from $2$. Is there any result on the existence of a point $y\in Y$ such that $X_y = ...

2
votes

1
answer

172
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### Preimage by birational maps

I am looking for an example (I guess that in complex projective space $\mathbb{P}^{n}$ is good) such that satisfy the following condition (in non trivial case, for this assume $X \neq \tilde{X}$):
Let ...

1
vote

1
answer

146
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### Local discriminant variety

I'm looking for good (as simple as it is possible) reference for the local discriminant variety.
I need it in the following situation:
I have an unfolding $F: (\mathbb{K}^n \times \mathbb{K}^p, 0) \to ...

2
votes

0
answers

131
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### plumbing description of resolution of ADE singularities

Let $G$ be a finite subgroup of $SU(2)$ and consider the quotient of the unit ball $B\subset \mathbb{C}^{2}$ by $G$. The result, denoted by $V$, has a boundary $S^{3}/G$ and has an ADE singularity at $...

6
votes

2
answers

475
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### Smooth complete intersections

Let $X_{2,3}\subset\mathbb{P}^n$, with $n\geq 5$, be a complete intersection of a quadric $X_2$ and a cubic $X_3$ containing a $2$-plane $H$. Assume $X_2$ and $X_3$ to be general among the ...

2
votes

1
answer

152
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### When is the singularity of a semi-normal variety a double point singularity

Let $X$ be a semi-normal projective variety and $p: \widetilde{X} \to X$ be the normalization. Suppose that $\widetilde{X}$ is smooth and there exists two smooth divisors $D_1, D_2 \subset \widetilde{...

0
votes

0
answers

37
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### Is there a treatment to relate the Multiple scale analysis or scale separation to the usage of the CFT especially in the perturbation?

In the recent years one started to think weather the abstract group treatment to the Multiple-scale analysis and the scale separation could be uniformly obtained. Especially, one attempted to search ...

7
votes

1
answer

259
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### Orbifolds are Thom-Mather stratified spaces

Where can I find a proof of (or if it is even true) that an (effective) orbifold is a Thom-Mather stratified space?
edit: after some search, I found the proof should be contained in either
GIBSON, C....

0
votes

0
answers

187
views

### Singularity of inverse exponential integral function

The exponential integral function is defined by
$$ Ei(z) = \int_{-\infty}^z dw \frac{e^w}{w} \,.$$
Away from the negative real axis the exponential integral function has a Taylor series about $z=0$:
$$...

4
votes

0
answers

71
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### Possible number of zeros of a stable perturbation of a germ $(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$

Let $f:(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$ be an analytic germ. Assume that it has isolated zero at 0, that is, $f^{-1}(0)=\{0\}$, what is more, assume that the dimension of the local algebra* $Q(...

1
vote

0
answers

76
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### Discrepancies and multiplicity of rational singularity

Let $(X,x)$ be a rational normal surface singularity having multiplicity $m$ (for example $(-Z)^{2}$, where $Z$ is the fundamental cycle). Suppose its discrepancies are all $\ge -1+\frac{1}{k}$ for a $...

5
votes

1
answer

311
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### Is there a, in depth, classification of branch points in complex analysis?

Disclaimer: This question was originally posted in math.stackexchange.com and, after 30 days with no answers, I followed the instructions of this topic.
In complex analysis we have well known results ...

4
votes

2
answers

582
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### Are Du Val singularities smoothable?

I am interested in when a Du Val surface singularity is smoothable. By Du Val singularity, I mean (the germ of a) isolated double point surface singularity admitting a resolution by blowups of ...

1
vote

1
answer

203
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### Singularities of Chow varieties

Let $X$ be a smooth complex projective variety. The Chow variety of degree $d$, $r$ dimensional subvarieties is denoted by $C_{d,r}(X)$. The Chow variety can have many topologically connected ...

7
votes

1
answer

342
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### Gorenstein varieties: why the two definitions are equivalent?

There are two definitions of Gorenstein singularities
in the literature. Using Grothendieck's (or Serre's) duality, one
defines the "dualizing sheaf" an object $\hat K_M$ of derived category
...

1
vote

0
answers

22
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### What is the relation between the different codimensions (e.g. left-right, contact) of map germs?

I would like to clarify the relations between the different codimension conceptions of map germs. I studied mostly from the new book of Mond and Nuno-Ballesteros, another source is Wall. Most of my ...

4
votes

1
answer

183
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### Is the pull-back of canonical sheaf invertible (modulo torsion)?

Let $X$ be a $\mathbb{Q}$-Gorenstein (isolated) singularity of dimension at least $2$ and $f:Y \to X$ be a resolution of singularities. In this case the canonical sheaf $K_X$ is not necessarily ...

3
votes

1
answer

145
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### Existence of terminal $3$-fold flips

Does there exist a terminal $3$-fold $X$ with a curve $C\subset X$ such that $K_X\cdot C < 0$ admitting a Mori flip $X\dashrightarrow Y$, flipping $C$ to a curve $C'\subset Y$, where the singular ...

1
vote

0
answers

43
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### Real (non-complex) Du Val singularities for quartics of high global Milnor number

I posted this in MathStackexchange and was advised to come here. As I am only looking for examples, I didn't feel MathOverflow was necessary.
I am looking for examples of specific quartic projective ...

11
votes

1
answer

429
views

### Does a resolution of a rational singularity have rationally connected fibers?

A rational singularity is a singularity of a
complex variety $X$ such that for any
resolution $\pi:\; \tilde X\rightarrow X$ the
higher direct images $R^i\pi_*(O_{\tilde X})$
vanish for all $i>0$. ...

3
votes

0
answers

100
views

### Tensor product by the canonical module preserves Cohen-Macaulayness

Let $X$ be a $\mathbb{Q}$-Gorenstein variety of dimension at least $2$. Suppose that $X$ is normal and Cohen-Macaulay with at worst isolated singularities. Let $F$ be a maximal Cohen-Macaulay $\...

5
votes

0
answers

76
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### forms on singular spaces that can be integrated on an LCI

I'd like to characterize rational $k$-forms on a singular scheme $X$ which can be integrated on any (real) bounded submanifold that is locally LCI in $X$ in a real sense (i.e., which is the real ...

2
votes

0
answers

118
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### Why does nearby cycles of a local system on $\mathbb{G}_m$ have same monodromy as local system?

Apologies if this belongs on MSE, but none of the tags I wanted existed, so I took it as a hint to post on MO.
Edit: here is my definition of nearby cycles. Suppose $X$ is a complex analytic space ...

4
votes

0
answers

121
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### How to judge whether an orbifold is good

My own case comes from dynamic system on compact complex manifolds. To be precise, let $M$ be a compact complex 3-dimentional manifold, $W^c$ a holomorphic foliation of M with 1-dimentional uniformly ...

4
votes

1
answer

190
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### Does the quotient of a variety with log terminal singularities also have log terminal singularities?

Boutot's theorem says that if $X$ is a variety over a field of characteristic 0 with rational singularities, and if $G$ is a reductive group acting on $X$, then the quotient $X/G$ has rational ...

4
votes

1
answer

190
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### Is pseudo-rationality preserved by etale morphisms?

Let $f: Y \to X$ be an etale morphism of schemes.
If $X$ has pseudo-rational singularities then does $Y$ also have pseudo-rational singularities?
For the definition of pseudo-rational see, for ...

4
votes

2
answers

171
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### Newton polygon notation for algebraic surface singularities

In various sources (e.g. here, Theorem 1.1 and here, Theorem 2.1 (3)), a certain notation which uses a fraction followed by a tuple is used to describe surface singularities. For example, the first ...

3
votes

0
answers

116
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### How to distinguish the singularities on moduli space?

Let me start with concrete examples. Let $X$ be a smooth special Gushel-Mukai threefold and $\mathcal{C}(X)$ be its honest Fano surface of conics, it has two irreducible components $\mathcal{C}(X)=\...

6
votes

1
answer

240
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### How to solve the following ODE with a parameter?

I am considering the following ODE
\begin{equation}
\begin{split}
&\frac{d^2}{dy^2}u + \frac{\alpha}{(1+y^2)^{\frac{r}{2}}}u = \delta(y)\\
&\lim_{|y|\to \infty}u(y) = 0.
\end{split}
\end{...

8
votes

0
answers

204
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### A criterion for rational singularities in mixed characteristic

Let $R$ be a mixed characteristic discrete valuation ring with perfect residue field and $f:X \to \mathrm{Spec}(R)$ a flat proper morphism.
If the generic fibre of $f$ is smooth and the special fibre ...

3
votes

0
answers

142
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### Relationship between exact symplectomorphisms and Hamiltonian diffeomorphisms (in the context of Milnor fibers)

Here is a preamble/setup. Suppose we have a symplectic manifold $(M,d\lambda)$ with an exact symplectic form. By Stokes theorem, $M$ must have nonempty boundary. An exact symplectomorphism $\phi:M \to ...

2
votes

0
answers

164
views

### Monodromy group of the Milnor fiber of an ADE surface singularity

Let $X$ be a hypersurfaces in $\mathbb C^3$ defined by $f(x,y,z)=0$ with an ADE type singularity at $0$. Denote $\mu$ the Milnor number of the singularity.
On the one hand, we can fit $(X,0)$ into the ...

3
votes

1
answer

135
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### A convolution type singular integral operator with log

Define a convolution type operator $T_m$ by
$$T_m(f) = p.v.\int_\mathbb{R}f(x-y)\frac{\log^m|y|}{y}dy.$$ Here $m\ge0$ is an integer.
Consider $f \in H^s (s > 0)$ which is the usual Sobolev space. ...