# Questions tagged [singularity-theory]

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

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### Krull dimension of the smooth locus

Let $R$ be a normal complete local domain of dimension $n \geq 4$. Does there exist a prime ideal $\mathfrak{p}$ of height $\dim(R) - 1$ such that $R_{\mathfrak{p}}$ is a regular local ring? In ...

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### Segre embedding and intersections by hyperplanes

Consider the Segre embedding $$ \mathbb{P}^2 \times \mathbb{P}^2 \to \mathbb{P}^8.$$ Denote by $V$ the image of the Segre embedding and by $B$ the locus of triples $(H_1, H_2, H_3)$ with $H_i \in H^0(\...

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### Normal forms of ADE singularities

Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms?
Does a similar ...

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### Derived category and resolution of singularities

Let $(X,x)$ be an isolated, Gorenstein singularity of dimension at least $2$ and $f: Y \to X$ be a resolution of singularities. Let $E_1, E_2$ be two distinct irreducible components of the exceptional ...

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### What are algebroid curves/branches and their value semigroup?

In “The moduli problem for plane branches”, by O. Zariski, the author defines a plane branch as an irreducible element $f \in \mathbb C[[x,y]]$. In the more recent article "The semigroup of a ...

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### Geometric interpretations of $A_k$ singularities on plane curves

Definition: A real smooth analytic function $f$ defined on some neighborhood of $t_0$ is said to have an $A_k$ singularity at $t_0$ if and only if $$ f'(t_0) = f''(t_0) = \dots = f^{(k)}(t_0) = 0$$ ...

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### Modality of a point under a Lie group action

Let $X$ be a smooth manifold and $G$ a Lie group acting on it. V. I. Arnold defines the modality of a point $x\in X$ as follows [1] (see also [2]):
We say that a point $x$ has modality $m$ (under the ...

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### How to determine the singlarity type (up to local analytic isomorphism) of a hypersurface surface singularity

Given a polynomial f(x,y,z), it defines a hypersurface in $\mathbb C^3$. I guess there is a classification of hypersurface singularity like Arnold normal form. I wonder given an explicite example of f,...

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### Apparent singularities and non Fuchsian regular points

I am considering the following function of $z$ on the Riemann sphere:
$$
J(z) = \int_\Delta (L_0+z L_1)^a D^b d^nx
$$
where
$\Delta \in H_n\big(\Bbb{CP}^n\setminus\{L(x)=0\},S\big)$, $S$ being the ...

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### Deformation to normal cone of the exception divisor of a log-resolution

I am reading the paper Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink due to G. Guibert, F. Loeser, and M. Merle. The main tool, like a lot of papers in ...

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### Definition of canonical pair

Let $(X,D)$ be a pair and $f:Y\rightarrow X$ a log resolution. Write
$$
K_Y + \widetilde{D} = f^{*}(K_X) + \sum_{i}a_iE_i
$$
where $\widetilde{D}$ is the strict transform of $D$. I found the following ...

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### An interpolation question for complex analytic hypervarieties

Suppose $D^{n} \subset \mathbb{C}^{n}$ is a small neighborhood of the origin, and $f, g: D^{n} \rightarrow \mathbb{C}$ are holomorphic functions. Moreover, $g$ is an arbitrarily small (i.e. $| f - g|_{...

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### Induced resolution of singularities

I am not a specialist in singularity theory but currently I have to touch resolution of singularities and I'd like to know whether I have understood Hironaka's theorem correctly.
Let $k$ be a field of ...

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### Converse of transfer theorem : does asymptotic behaviour of coefficients describe the singularity?

I'm interested in Singularity Analysis after reading this Math.Stack post. In the "Analytic Combinatorics" book by P. Flajolet and R. Sedgewick, at p.390, the following transfer theorem (...

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### Two notions of singular support?

Arinkin-Gaitsgory have defined the notion of singular support for any quasismooth $Y$
$$\text{SS}(\mathcal{F})\ \subseteq\ \text{Sing}(Y)$$
and $\mathcal{F}$ any ind-coherent sheaf, where $\text{Sing}(...

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### Example of nontrivial families of isolated singularities with constant Milnor number

In Lê-Ramanujam's paper The invariance of Milnor’s number implies the invariance of the topological
type, they prove what the title says for families with isolated singularities and constant Milnor ...

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### Vorticity equation for incompressible 2D fluid dynamics [closed]

I want to ask what advantage of using vorticity equations in fluid dynamics.
Does it help to find large curls? Does it have singularities connected to presence of curls?

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### Trace formula for monodromy of Milnor fibrations

I am reading the paper A. Campo, Le nombre de Lefschetz d'une monodromie but I am stuck at several points, hope that someone can help me.
Let $P:\mathbb{C}^{n+1} \longrightarrow \mathbb{C}$ be a germ ...

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### Every locally free sheaf is Cohen-Macaulay on complex variety with at canonical singualities?

Suppose $X$ is a normal complex space with at most singularities, can we say any locally free sheaf on it is Cohen-Macauly?
Recall that a coherent sheaf $\mathcal{F}$ over $X$ is called (maximal) ...

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### Understand the proof that rational resolution is independent of the resolution

EDIT. Karl Schwede has given a different approach to solve this question, which is very clear. However, I still want to figure out how to deal with this problem along Ein's line, and waiting for the ...

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### Discrepancy of a divisor over a different model

I also asked this question on MathStackExchange but receive no answers.
I'm reading Koll'ar and Mori's book about singularity theory. They state the following lemma without proof:
Lemma 2.30. Let $f:...

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### Curvature explosion and metric landmark stability

$\newcommand{\prin}{\mathrm{prin}}$Context: Let $S$ be the unit sphere in some finite dimensional vector space $V$. Given a connected compact Lie group real representation $\rho:G\rightarrow O(V)$, ...

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### Intersection multiplicity via parametrization in general

My question is a generalization of Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide.
Take two complex space germs $(A, 0)=V(I_A)$ of dimension $...

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### Bishop's compactness theorem and convergence of analytic subset

Let $V_i$ be a sequence of $k$ dimensional analytic subsets in $\mathbb C^N$. Suppose that the volume of $V_i$ is uniformly bounded, then Bishop's compactness theorem says that $V_i$ will convergence ...

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### Singularities of contractions of extremal faces

Let $(X, \Delta)$ be a (projective) klt pair (say over $\mathbb{C}$, but I am also interested in fields of positive characteristic) and $f: X \to Z$ the contraction associated to a $(K_X + \Delta)$-...

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### When a strictly positive log pluriharmonic function $g$ is equal to the norm of holomorphic function?

Suppose $V$ is a local analytic variety (singular). Suppose $g$ a strictly positive log pluriharmonic function on $V$, i.e. $\log g$ is pluriharmonic. I wonder when $g=|f|^2$, where $f$ is a ...

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### Codimension of cusp singularities in the space of 2-jets

In trying to prove Cerf's theorem about homotopies between Morse-functions I ended up thinking about the following problem.
For $n>2$, $a= (a_{i,j})\in GL(n-2)$, we define the polynomial map $C_a:\...

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### Order of a germ along curves

There is a very basic invariant of $\mathcal{C}^{\infty}$-germs $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^k, 0) $ ($n \leq k$), which I call ''maximal order along curves''. I would like to interested ...

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### Is this quotient of $\mathbb{C}^{m+1}$ by $U(1)$ only "nice" for $m=1$?

Let $V^{m+1} = \mathbb{C}^{m+1}$ and let $U(1)$ act on it by its diagonal representation, so that really, it is just like scalar multiplication by a unit modulus complex number.
I am interested in the ...

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### Does anyone know an example of a non-singular, globally $ F $-regular variety $ X $ for which generic smoothness does not hold?

Let us denote the Frobenius endomorphism of a variety $ X $ by $ F $. A variety $ X $ over a field $ k $ of positive characteristic is globally $ F $-regular if for every effective Weil divisor $ D $,...

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### Singular Del Pezzo of degree 2

Throughout, singular Del Pezzo means a surface with only isolated singularities and ample anti-canonical divisor.
Suppose $X$ is a singular Del Pezzo of degree 2 over a field $k$ where $\text{char}(k)\...

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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 $\...

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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 ...

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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, ...

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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 ...

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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.
...

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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 ...

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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 ...

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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 $m$...

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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 ...

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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 ...

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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 $\...

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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 ...

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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 ...

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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&...

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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\...

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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.

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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 $G=SL(n+1)$ ...

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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 = ...

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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 ...