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Questions tagged [singularity-theory]

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

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Moduli space of curves away from singular subsets

Recently, I'm interested in the moduli spaces of curves in a (possibly noncompact) complex orbifold (resp. symplectic and almost complex) away from singularities. More specifically, I'm interested in ...
ChoMedit's user avatar
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Examples of small resolutions in dimension 4 and higher

I have seen numerous examples of 3-folds admitting small resolutions. Are there similar examples in higher dimension? In particular, I am looking for examples of singular varieties of dimension $4$ ...
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Theories of manifolds w/ extra structure and singularities

Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
Will Sawin's user avatar
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Any "motive"(-like) theory which can catch that cusp $y^2=x^3$ (and similar) are non-trivial?

Consider cusp $y^2=x^3$ which can also be described as $k[z]~without~z$ , taking $x=z^3,y=z^2$. Algebraically its $Spec$ is quite different from $k$. For example: it has plenty non-trivial "line-...
Alexander Chervov's user avatar
1 vote
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Solutions to ODE/SDE with singular coefficients $dX_t = -X_t/t \, dt + g\,dW_t$

I encountered a question regarding the solutions to SDEs with singular drifts. I searched the literature but had a hard time figuring out the intuition behind these analytic results assuming different ...
Yifan's user avatar
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42 views

Perturbation method for time-periodic singular system of ODEs

I am studying a problem arising in physics, and I managed to simplify it to a differential system (initial value problem) of the form: $$ \begin{cases} \dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
squille's user avatar
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Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$

In my research I encountered automorphisms of the ring of convergent power series $$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$ which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
red_trumpet's user avatar
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Is any singularity a subgerm of $(\mathbb{C}^n, 0)$?

I am studying singularity theory. I have often come across, in the literature, the sentence which says "let $(X,0) \subset (\mathbb{C}^n,0)$ be a singularity". Here a singularity is a ...
Math1016's user avatar
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Formal neighborhood of isolated singularity via DAG

I work over a field of characteristic $0$, denoted $k$. Let $f:\mathbf{A}^{d+1}\rightarrow\mathbf{A}^{1}$ have an isolated singularity at $0$, and let $\widehat{Z}$ denote the formal neighborhood of $...
EBz's user avatar
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Continuous invariants of singularities in the Thom-Mather theory of deformations

I have been reading through Arnold et al.'s Singularities of differentiable maps to have an understanding on Arnold's theory of deformations of wave fronts. His theory is similar to the Thom-Mather ...
Cuspidal Coffee's user avatar
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Characterization of the Picard's condition for integral equation

Picard's condition (Thm. 15.18, Kress et al. 1989) is essential to study the existence of a solution of a Fredholm integral equation of the first kind. Specifically, consider (the univariate case) the ...
Mingzhou Liu's user avatar
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213 views

Is the Taylor map continuous?

(Skip to the bolded theorem below for my question, if you'd like) Some context on asymptotic expansions and the Taylor map In the setting of irregular singularities of meromorphic connections on the ...
Brian Hepler's user avatar
2 votes
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174 views

A question about the regularity of the Schrödinger equation

While reading the article [1], I noticed I don't understand part of the proof of regularity. For the Schrödinger eigenvalue problem, \begin{cases} -\Delta u+Vu=\lambda u, &\text{in } \Omega \\ \...
Du Xin's user avatar
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Why the following quasi isomorphism implies the morphism to be a resolution (a step in the paper A characterization of rational singularities)

I was reading the paper A Characterization of Rational Singularities by Professor Kovács. The main theorem is stated as follows: THEOREM 1. Let $\phi: Y \rightarrow X$ be a morphism of varieties over ...
yi li's user avatar
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Is there a way to solve this integral on the sphere explicitly?

Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that $k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral $$f(y):=\int_{\...
Medo's user avatar
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Solvability of derivation Lie algebras of local finite-dimensional commutative algebras

Let $A$ be a finite-dimensional local commutative algebra (with one) over a characteristic zero field $k$. Is it true that the Lie algebra $\operatorname{Der}_k(A)$ of $k$-derivations of $A$ is ...
მამუკა ჯიბლაძე's user avatar
1 vote
0 answers
109 views

Contracting a family of rational curves in a Calabi Yau threefold

Suppose we have a Calabi-Yau 3-fold $X$ (not necessarily compact, over $\mathbb{C}$) that contains a ruled surface over a smooth curve $C$ of genus $g$. I am using a strong definition of a ruled ...
Sungwoo's user avatar
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4 votes
2 answers
519 views

Are singularities of complex varieties captured by topology?

Let $X \subseteq \mathbb{C}^n$ be an affine complex algebraic variety, with a singularity at some point $x.$ Let $U \subseteq \mathbb{C}^n$ be an open set containing $x$. Can we determine if $x$ is a ...
Michael Barz's user avatar
1 vote
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92 views

Poisson summation for solutions of the Burgers equation in the form 1/x

Long story short: I'm looking for a good way of showing that the Fourier transform of $1/x$ is a sign function. Motivation and why this has been a problem: I'm dealing with an equation similar to the ...
Rafael's user avatar
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3 votes
0 answers
112 views

Proof of rationality of roots of Bernstein polynomial à la Lê

Lê Dũng Tráng has a paper "The Geometry of the Monodromy Theorem" (MR0541020 https://mathscinet.ams.org/mathscinet/article?mr=541020 for reference). It's a nice paper, and in it Lê gives a ...
Michael Barz's user avatar
3 votes
1 answer
362 views

What does the Jacobian of a vector field at an equilibrium tell you about local behavior of integral curves when the Jacobian is not a stable?

I have a soft question regarding the Jacobian of vector fields and isolated equilibria, and what they imply about local behavior of nearby integral curves near. Let $V:U \subset_{open} \mathbb{R}^n \...
Spencer Kraisler's user avatar
3 votes
2 answers
379 views

Cohen-Macaulay Representations

I came across the book "Cohen-Macaulay Representations" by Graham J. Leuschke and Roger Wiegand, and now I'm wondering if this is an active area of research. If yes, then what are some of ...
It'sMe's user avatar
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2 votes
0 answers
91 views

Non-compactness on Penrose singularity

I've been studying singularities in GR, and (obviously), came across PST. Let us state it as the following: Let $(M, g)$ be a connected globally hyperbolic spacetime with a noncompact Cauchy ...
Johann Wagner's user avatar
1 vote
1 answer
118 views

Singularities of fibrations in conics

Consider a rank two vector bundle $E = \mathcal{O}(a)\oplus \mathcal{O}(b)\oplus \mathcal{O}(c)$ over $\mathbb{P}^1$. Fix coordinates $u_0,u_1$ on the base $\mathbb{P}^1$ and $v_0,v_1,v_2$ on the ...
Puzzled's user avatar
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2 votes
1 answer
358 views

What is the ideal of hypersurfaces singular at a given irreducible variety?

Let $X\subseteq \mathbb{P}^n$ be a closed irreducible subvariety, with vanishing ideal $I(X)\subseteq k[x_0,\ldots,x_n]$, where $k$ is the ground field, assumed to be algebraically closed. Let $F\in k[...
Jérémy Blanc's user avatar
1 vote
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118 views

Multiplicity of a singular point in a Schubert-like variety

Let us fix the base field to be the field of complex numbers (Maybe it's not quite important). Recall the following definition. Let $X$ be a quasi-projective variety, singular at a point $x$. Let $C_{...
Pène Papin's user avatar
2 votes
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125 views

Making a continuous function into embedding by adding additional dimension

While doing my researches, I encountered the following problem. Let $f:[0,1]^n\rightarrow \mathbb{R}^{n+k}$ be an arbitrary continuous function. I want to make this function an embedding by perturbing ...
GHG's user avatar
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2 votes
1 answer
303 views

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 ...
Shravan Patankar's user avatar
3 votes
1 answer
284 views

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(\...
user45397's user avatar
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2 votes
1 answer
267 views

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 ...
H U's user avatar
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2 votes
0 answers
107 views

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 ...
user45397's user avatar
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1 vote
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120 views

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 ...
Lucas Henrique's user avatar
3 votes
0 answers
111 views

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 $$ f'(t_0) = f''(t_0) = \dots = f^{(k)}(t_0) = 0$$ ...
Arthur Queiroz Moura's user avatar
3 votes
0 answers
54 views

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 ...
igorf's user avatar
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0 answers
69 views

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,...
xin fu's user avatar
  • 613
2 votes
0 answers
57 views

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 ...
0x11111's user avatar
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2 votes
0 answers
98 views

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 ...
Alexey Do's user avatar
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-1 votes
1 answer
176 views

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 ...
Puzzled's user avatar
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1 vote
1 answer
185 views

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 ...
Alexey Do's user avatar
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1 vote
0 answers
106 views

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 (...
Desura's user avatar
  • 221
4 votes
1 answer
408 views

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}(...
Pulcinella's user avatar
  • 5,555
1 vote
0 answers
78 views

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 ...
inkievoyd's user avatar
  • 508
0 votes
1 answer
158 views

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?
Dragomir's user avatar
2 votes
0 answers
170 views

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 ...
Alexey Do's user avatar
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0 votes
0 answers
102 views

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) ...
Invariance's user avatar
5 votes
1 answer
352 views

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 ...
Invariance's user avatar
1 vote
0 answers
111 views

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:...
Hydrogen's user avatar
  • 313
3 votes
0 answers
118 views

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 $\rho:G\rightarrow O(V)$, ...
miniii's user avatar
  • 59
1 vote
0 answers
136 views

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 $...
Gergo Pinter's user avatar
2 votes
1 answer
206 views

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 ...
xin fu's user avatar
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