Questions tagged [singularity-theory]

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

Filter by
Sorted by
Tagged with
4 votes
0 answers
84 views

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 $\...
user avatar
  • 301
2 votes
0 answers
66 views

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 ...
user avatar
1 vote
1 answer
83 views

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, ...
user avatar
2 votes
1 answer
93 views

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 ...
user avatar
  • 355
4 votes
1 answer
126 views

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. ...
user avatar
  • 355
2 votes
0 answers
85 views

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 ...
user avatar
1 vote
0 answers
127 views

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 ...
user avatar
2 votes
1 answer
165 views

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 $m$...
user avatar
2 votes
0 answers
53 views

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 ...
user avatar
-1 votes
1 answer
82 views

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 ...
user avatar
  • 11
2 votes
0 answers
90 views

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 $\...
user avatar
  • 1,940
4 votes
0 answers
135 views

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 ...
user avatar
4 votes
1 answer
163 views

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 ...
user avatar
  • 1,940
1 vote
0 answers
56 views

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&...
user avatar
  • 720
2 votes
0 answers
80 views

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\...
user avatar
  • 21
3 votes
1 answer
94 views

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.
user avatar
3 votes
0 answers
139 views

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 $G=SL(n+1)$ ...
user avatar
4 votes
2 answers
437 views

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 = ...
user avatar
2 votes
1 answer
172 views

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 ...
user avatar
  • 131
1 vote
1 answer
146 views

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 ...
user avatar
2 votes
0 answers
131 views

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 $...
user avatar
  • 979
6 votes
2 answers
475 views

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 ...
user avatar
  • 301
2 votes
1 answer
152 views

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{...
user avatar
  • 1,533
0 votes
0 answers
37 views

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 ...
user avatar
7 votes
1 answer
259 views

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....
user avatar
  • 751
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 $z=0$: $$...
user avatar
4 votes
0 answers
71 views

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(...
user avatar
1 vote
0 answers
76 views

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 $...
user avatar
5 votes
1 answer
311 views

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 ...
user avatar
4 votes
2 answers
582 views

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 ...
user avatar
  • 205
1 vote
1 answer
203 views

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 ...
user avatar
  • 4,926
7 votes
1 answer
342 views

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 ...
user avatar
1 vote
0 answers
22 views

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 ...
user avatar
4 votes
1 answer
183 views

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 ...
user avatar
  • 1,939
3 votes
1 answer
145 views

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 ...
user avatar
  • 301
1 vote
0 answers
43 views

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 ...
user avatar
  • 111
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 $i>0$. ...
user avatar
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 $\...
user avatar
  • 1,939
5 votes
0 answers
76 views

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 ...
user avatar
2 votes
0 answers
118 views

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 ...
user avatar
  • 273
4 votes
0 answers
121 views

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 ...
user avatar
4 votes
1 answer
190 views

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 ...
user avatar
  • 93
4 votes
1 answer
190 views

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 ...
user avatar
  • 10k
4 votes
2 answers
171 views

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 ...
user avatar
3 votes
0 answers
116 views

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)=\...
user avatar
  • 1,516
6 votes
1 answer
240 views

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{...
user avatar
  • 883
8 votes
0 answers
204 views

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 ...
user avatar
  • 10k
3 votes
0 answers
142 views

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 ...
user avatar
  • 438
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 ...
user avatar
  • 1,013
3 votes
1 answer
135 views

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. ...
user avatar
  • 883

1
2 3 4 5
10