Questions tagged [singularity-theory]
Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.
554 questions
8
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1
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Fibers of generic smooth maps between manifolds of equal dimension
I have heard that the following is a "well-known"
Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite ...
- 501
2
votes
1
answer
202
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Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4.
Near the end of section 3, ...
- 129
1
vote
0
answers
85
views
Projection from a point and singularity
Let $X \subset \mathbb{P}^n$ be a hypersurface with $n \ge 3$. Let $x \in X$ be a closed point. Consider the map given by projection from $x$:
$$\phi: X \dashrightarrow \mathbb{P}^{n-1}$$
Suppose that ...
- 1,070
1
vote
0
answers
57
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Discrepancy of general element of linear system
Let $X$ be a normal scheme and $|D|$ a linear system on $X$.
In "Singularity of Minimal Model Program" by Janos kollar p249, it says,
If $X$ is a variety over $\mathbb{C}$, and $E_j$ ...
- 328
2
votes
0
answers
109
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Punctured neighbourhood of quotient singularity is not simply connected?
Let $X$ be a variety (irreducible, normal) over a field $k$ which is algebraically closed with characteristic $0$. Suppose that $X$ has only one singular point $p\in X$, so $Y:=X\setminus p$ is smooth....
- 131
1
vote
0
answers
35
views
Does every holomorphic map admit a stratified submersion?
Given a map (germ) $g:(\mathbb{C}^{n+k},0)\rightarrow (\mathbb{C}^k,0)$, are there stratifications that make it a stratified submersion?
By stratified submersion I mean a map that has stratifications ...
- 333
0
votes
0
answers
34
views
Bôcher's theorem for singularities on the boundary
Let $\Omega\subset\mathbb{R}^2$ be connected, open, bounded, and smooth. Suppose that $u\in C^0(\bar \Omega\setminus \{0\})\cap C^2(\Omega\setminus\{0\})$ is harmonic and positive in $\Omega$.
If $0\...
- 308
2
votes
1
answer
117
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Blow up of terminal singularity and canonical singularity
A normal singularity $(X,x)$ over a field $k$ is terminal (resp. canonical) if
$(i)$
it it is $\mathbb{Q}$-Gorenstein. and
$(ii)$For any resolution of singularity $F:Y\rightarrow X$,
$K_Y-f^*K_X>...
- 328
5
votes
1
answer
469
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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-...
- 24.9k
7
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0
answers
277
views
Is every normalization a blowup?
I asked this at math.stackexchange, but received no reply.
Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example.
...
2
votes
0
answers
108
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Finiteness of rational double point
Let $(R,\mathfrak{m
})$ be a three dimensional complete local ring over a field $k$ of arbitrary characteristic and let $f\in R$ and $R/f$ is a rational double point.
My question is as follows.
Are ...
- 328
0
votes
0
answers
78
views
Log resolution and a divisor of pullback of function
Let $(X,x)$ be a three fold singularity
$m_{X,x}$ a ideal sheaf correspoinding to $x$.
$\sigma:Y_1\rightarrow X$ blow up at by $m_{X,x}$
$\phi:Y\rightarrow Y_1$ resolution of $Y_1$
Set $f:=\phi*\sigma$...
- 328
1
vote
0
answers
42
views
Concerning the definition of a class of functions introduced by Nilsson
In the paper 'Some growth and ramification properties of certain integrals on algebraic manifolds' by Nilsson we find the following definitions:
My question is how does one prove the remark "It ...
- 360
2
votes
0
answers
46
views
Smooth vs. topological: foliation into closures of orbits
Consider a (partial) map $f\colon X → X$ and the maximal closures of orbits of $f$ (i.e., closures of orbits which are not contained in larger closures of orbits). Assume that $X$ is foliated into ...
- 455
1
vote
1
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151
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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 $4$ ...
- 2,323
7
votes
0
answers
486
views
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 ...
- 149k
1
vote
0
answers
62
views
About the definition of cDV singularity
M. Reid defines cDV singularity as follow in his paper "CANONICAL 3-FOLDS"
A point $p\in X$ of a 3-fold is called a compound Du Val point if for some section H throgh $P$, $P\in H$ is a Du ...
- 328
1
vote
0
answers
54
views
Estimating the growth rate around singular points of the analytic continuation of functions of Nilsson class defined by an integral
In Lemma 8 of the paper "Constant terms in powers of a Laurent polynomial" (by J.J. Duistermaat and Wilberd van der Kallen) the exponent $\alpha$ in the asymptotic expansion of a function of ...
- 360
4
votes
2
answers
588
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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 ...
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6
votes
0
answers
200
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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 ...
- 1,286
1
vote
0
answers
51
views
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 ...
- 285
1
vote
0
answers
159
views
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 ...
- 73
2
votes
1
answer
138
views
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 ...
- 18.4k
6
votes
0
answers
220
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 ...
- 370
0
votes
0
answers
121
views
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 ...
- 369
3
votes
0
answers
57
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:
$$
\begin{cases}
\dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
- 131
5
votes
0
answers
243
views
Is there a way to solve this integral on the sphere explicitly?
Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that
$k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral
$$f(y):=\int_{\...
- 852
2
votes
1
answer
162
views
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 ...
- 225
2
votes
0
answers
193
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 \\
\...
- 31
3
votes
1
answer
402
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 \...
- 518
2
votes
1
answer
387
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[...
- 7,722
2
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0
answers
91
views
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 $...
- 121
3
votes
2
answers
396
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 ...
- 839
2
votes
0
answers
77
views
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 ...
1
vote
0
answers
123
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 ...
- 71
1
vote
0
answers
52
views
Making sense of constant scalar curvature metric horns
Suppose we have a compact oriented surface $S$ and we remove a point $p$ on it. We could consider a neighboorhood $U$ of the puncture $p$, so that the points in this neighboorhood are described by ...
2
votes
1
answer
327
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 ...
1
vote
1
answer
136
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 ...
- 8,998
3
votes
1
answer
327
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(\...
- 2,323
2
votes
1
answer
304
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 ...
- 481
1
vote
0
answers
105
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 ...
- 93
1
vote
0
answers
45
views
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 ...
- 143
4
votes
1
answer
533
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}(...
- 5,711
2
votes
0
answers
128
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 ...
- 173
3
votes
0
answers
117
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 ...
- 813
1
vote
0
answers
128
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_{...
- 279
0
votes
1
answer
257
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 ...
- 8,998
1
vote
1
answer
241
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 ...
- 8,998
4
votes
0
answers
127
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 $$ f'(t_0) = f''(t_0) = \dots = f^{(k)}(t_0) = 0$$ ...
75
votes
4
answers
6k
views
When is a singular point of a variety ($\mathcal{C}^\infty$-) 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 ...
- 47.5k