Questions tagged [singularity-theory]
Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.
554 questions
12
votes
0
answers
443
views
What is the motivation for a Frobenius manifold?
A Frobenius manifold is a type of manifolds with extra structure.
The main examples are quantum cohomology (viewed as a space itself), GBV algebras, the ``Saito'' examples arising from singularities (...
- 5,711
3
votes
1
answer
199
views
Is it possible that $\int |f^2(z)|^{t+1}(P\bar{P})\phi(z) dz=0$ for all compactly supported $\phi$?
When proving that the log-canonical threshold is minus the largest root of the Bernstein-Sato polynomial one considers the integral
$$\int |f^2(z)|^{t+1}(P\bar{P}(\phi(z))) dz,$$
where $P$ is a linear ...
- 2,788
15
votes
1
answer
1k
views
The homology groups of the smooth locus of a singular variety
Let $X$ be a complex irreducible variety and denote its smooth locus by $X^{smooth}$. I would like to know what can be said about the induced maps $H_i(X^{smooth};\mathbb{Q})\rightarrow H_i(X;\mathbb{...
- 3,599
2
votes
0
answers
191
views
Blowing up the zero section for "Chasse au Canard" (some new kind of geometric canards)
In this paper "Canard cycles and center manifolds" one encounters the blowing up of a non isolated set or manifold of singularities of a vector field or a singular foliation. This is a ...
- 366
1
vote
0
answers
219
views
Desingularization of the zero section of $TM$ as the manifold of singularities of the geodesic flow
However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities"...
- 366
2
votes
1
answer
1k
views
Essential singularity [closed]
In shaum's outline complex analysis,definition of essential point is:
An isolated singularity that is not pole or removable singularity is called essential singularity
Now in the same book there is an ...
18
votes
1
answer
830
views
Cohomology of real analytic coherent sheaves
Let $M$ be a real analytic variety
(if someone is concerned about distinction between
"real analytic spaces" and "real analytic varieties"
in real analytic geometry, let's assume that $M$
is both "...
- 9,237
5
votes
1
answer
322
views
Is identification of double points of an immersion smooth?
Let $f:M^m\to N^n$ be a generic map between smooth manifolds $n>m$. Depending on the pair $(m,n)$ generic maps will have a singular set of double points $\Sigma_2\subset M$.
Let $\phi:\Sigma_2\to \...
- 2,533
3
votes
1
answer
217
views
Weak Fano varieties and small transformations
A projective normal and $\mathbb{Q}$-factorial variety $X$ is said to be log Fano if there exists and effective divisor $D$ on $X$ such $-K_X-D$ is ample and the pair $(X,D)$ is klt.
Now, let $f:X\...
user168611
7
votes
1
answer
372
views
Non-example for Whitney (a) stratifications
Given a $C^1$ stratification $\mathscr{S}$ of a $C^1$ manifold $M$, we write $N^\ast \mathscr{S}$ for the union of conormals to the strata. The stratification is said to be Whitney (a) if $N^\ast \...
- 525
6
votes
2
answers
985
views
A paradox on the deformation of singularities
Setup: $\pi: \mathcal X \to C$ is a flat morphism from a germ of a smooth curve $(C, o)$. Suppose the special fiber $\pi^{-1}(o) := \mathcal X_o$ has a certain class of singularities, one wants to ...
- 3,472
8
votes
0
answers
349
views
Example of torsion differential forms
I am looking for an example of a normal affine variety $V$ over a perfect field $k$ such that the differentials $\Omega_{V/k}$ are not torsion free.
If normality is not required, an example is given ...
- 227
8
votes
1
answer
600
views
Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have distinct singular values?
$\newcommand{\SO}[1]{\text{SO}(#1)}$
$\newcommand{\dist}{\operatorname{dist}}$
Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth.
Set
$$...
- 6,741
0
votes
0
answers
135
views
On resolution of singularities over an Artin ring
For a locally noetherian scheme $X$, Grothendieck conjectured that if $X$ is quasi-excellent then there is a proper birational map $Y \to X$ s.t. $Y$ is regular.
We now fix an Artin ring $R$ whose ...
- 519
2
votes
0
answers
237
views
Singularity of L^1-solutions to elliptic PDEs on the puntured ball
Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^n$. Then it is true that if $u\in L^1(\mathbb{B})$ such that $\Delta(u)=0$ on $\mathbb{B}\backslash\{0\}$, then $\Delta(u)$, as a distribution on $\...
- 577
3
votes
1
answer
331
views
Smoothable $\mathbb{F}_p$-variety embeds in a regular scheme
Let $X$ be a proper geometrically integral $\mathbb{F}_p$-scheme.
Assume that $X$ is the special fiber of a proper flat $\mathbb{Z}_p$-scheme with a smooth generic fiber and that for each point $x\in ...
user158636
2
votes
0
answers
288
views
Torsion freeness of direct image of structure sheaf?
I have the following question.
Let $f:X\rightarrow Y$ be a surjective projective morphism between smooth projective varieties.
I learned that if $\dim Y=1$, then $R^if_*\mathcal O_X$ is torsion free $\...
- 623
9
votes
0
answers
148
views
Does every sequence of deformation of singularities eventually become equisingular?
Suppose we are over a field of characteristic zero and $f_i\colon X_i\to \mathrm{Spec}(R_i)$ $(i=1,2,\cdots)$ are flat families of singularities over DVRs. Assume that the generic fiber of $f_i$ is ...
- 131
7
votes
1
answer
231
views
Resolution graphs in the sense of Némethi
The following definitions are from lecture notes of Némethi. A surface singularity $(X,0)$ is defined by $$(X,0) = (\{ f_1 = \ldots = f_m=0 \}) \subset \mathbb (\mathbb{C}^n,0),$$ where $f_i : (\...
user150450
6
votes
1
answer
293
views
Riemann-Hurwitz for real maps
Let $S$ be a (compact, connected) Riemann surface of genus $g$ and $f: S\to \mathbb CP^1$ be a degree $d$ meromorphic function. Then the Riemann-Hurwitz formula tells us that the number of ...
- 14.3k
32
votes
3
answers
3k
views
Wanted: example of a non-algebraic singularity
Given a finitely generated $\def\CC{\mathbb C}\CC$-algebra $R$ and a $\CC$-point (maximal ideal) $p\in Spec(R)$, I define the singularity type of $p\in Spec(R)$ to be the isomorphism class of the ...
- 24.1k
8
votes
1
answer
496
views
Is canonical model always with canonical singularity
Let X be a smooth variety, take the proj of canonical ring of X and denote it by Y. Is Y always a canonical variety? I know it's true for general type variety. Thank in advance.
- 623
5
votes
0
answers
166
views
Normal singularities homeomorphic to a smooth space
I am looking for examples of normal complex spaces $X$ which locally around a singular point are homeomorphic to a smooth complex manifold.
The only example I know is a curve with a cusp, but this is ...
- 2,384
2
votes
0
answers
62
views
Whitney stratification of a $\mathbb C^*$-invariant hypersuface in $\mathbb C^n$
Let $H\subset \mathbb C^n$ be an irreducible hypersurface invariant under a diagonal $\mathbb C^*$-action with positive weights ($H$ is given by a quasi-homogeneous polynomial). Consider the Whitney ...
- 14.3k
7
votes
1
answer
770
views
Cohomology of tangent sheaf of a singular hypersurface
Let $X\subset\mathbb{P}^n$ be a hypersurface singular at finitely many points $p_i\in X$. We may assume that $X$ has ordinary singularities at the $p_i$'s.
Does there exists a formula, perhaps in ...
user91659
3
votes
0
answers
199
views
Divisorial contractions and singularities
I have a smooth $6$-fold $X\subset\mathbb{P}^n$ and a divisor $D\subset X$ cut out by a quadratic polynomial. I know that $D$ in singular along a smooth $3$-fold $Y\subset X$, and that if $Z$ is the ...
user125056
2
votes
1
answer
259
views
Scheme-theoretic image and delta-invariants
Let $(X,o)$ be an affine, isolated, normal, Gorenstein singularity. Let $f$ and $g$ be two morphisms from $\mbox{Spec}(\mathbb{C}[[t]])$ to $X$ (also known as formal arcs) such that the closed point ...
- 1,593
1
vote
0
answers
927
views
canonical divisor on singular curves with nodal point
What's the definition of canonical divisor(or whatever related concept) on singular curve with nodal point. More generally, what the definition of canonical divisor on a singular variety X, which is ...
- 623
5
votes
0
answers
494
views
ramification locus for finite morphism and Abhyankar's Lemma
I want to ask given a finite morphism between projective varieties $f:X\rightarrow Y$.
What is exactly the ramification locus $\Delta(X/Y)$. If $X$, $Y$, $f$ are smooth, then I can more or less ...
- 623
14
votes
5
answers
2k
views
Singular semi-Riemannian Geometry: usefulness and state of the art
My question has two parts, one concerning the state of the art of the subject, and the other the usefulness.
1. State of the art.
Can someone provide references reflecting the state of the art in ...
- 4,312
6
votes
0
answers
388
views
Globalization of Brieskorn-Grothendieck resolution
Brieskorn (1970) showed that for semiuniversal deformation of rational double points surface singularities $X\to S$, there is a finite base change $S'\to S$, such that the new family $f:X\times_{S}S'\...
- 1,803
4
votes
1
answer
312
views
Kähler-Einstein metrics on singular varieties
Let $X$ be a normal projective variety with klt singularities with numerically trivial canonical divisor $K_X$.
Does there always exist a Kähler-Einstein metric on $X$?
user58018
5
votes
1
answer
273
views
Singularities of curves that are moving
Let $k$ be an algebraically closed field, let $d\ge 2$ be an integer and let $f,g\in k[x,y,z]$ be two homogeneous polynomials of degree $d$ without common factor.
We want to know what are the ...
- 7,722
25
votes
1
answer
4k
views
Can the constant rank theorem for smooth manifolds be generalized to nonconstant rank?
The constant rank theorem says that
if $f\colon M→N$ is a smooth map whose rank equals some fixed $k≥0$ at any point of $M$, then, locally with respect to $M$ and $N$, the map $f$ assumes the easiest ...
- 37.8k
1
vote
0
answers
67
views
Robust features intuition?
The terminology robust features was introduced by Ian Porteous as they are features of a surface wich be followed when the surface is deformed. They capture important aspects of the surface geometry. ...
- 119
1
vote
0
answers
27
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 ...
- 73
6
votes
1
answer
1k
views
The singularity of the algebraic stack and the singularity of the coarse moduli space
It is possible that an algebraic stack is smooth while the coarse moduli space is not smooth. I want to know what is relationship between the singularity of the algebraic stack and that of its coarse ...
- 493
6
votes
1
answer
182
views
Can we approximate a vector field on the plane with non-vanishing vector fields in $W^{1,2}$?
Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated.
Does there exist a sequence of ...
- 6,741
4
votes
2
answers
2k
views
KLT singularities are quotient in codimension 2
I have read that if a variety $X$ has KLT singularities, then it has quotient singularities in codimension 2.
Do you know a proof (or where can I find a proof) of this?
- 1,150
1
vote
1
answer
125
views
Applying analytic coordinate changes to singular function germs [closed]
Suppose we are given a function germ
\begin{align}
f = \sum a_{ijk}x^iy^jz^k
\end{align}
such that $f\in \mathfrak{m}^2$, where $\mathfrak{m}$ is the ideal in $\mathbb{C}\{x,y,z\}$ of holomorphic ...
- 19
1
vote
1
answer
122
views
Tangent space to subspace of orbit in jet spaces
I consider a map germ $f: (\mathbb{R}^n,0) \to \mathbb{R} $ which is $k$-determined for some $k \in \mathbb N$, i.e. for all map germs $g: (\mathbb{R}^n,0) \to \mathbb{R} $ having the same $k$-jet as $...
- 113
1
vote
0
answers
245
views
Dual varieties and nodal sections
Let $X$ be a(n even dimensional) smooth complex projective variety in $\mathbb{P}^N$, and let $X^{\vee}$ be its dual variety; up to an higher degree Veronese embedding of $X$, I assume that $X^{\vee}$ ...
- 828
2
votes
1
answer
317
views
Singular locus of a linear system of hyperplane sections
Let $X\subset\mathbb{P}^N$ be a rational smooth projective irreducible non degenerated variety of dimension $n=\dim(X)$ and let $$\mathcal{H}=|\mathcal{O}_X(1) \otimes\mathcal{I}_{{p_1}^2,\dots,{p_l}^...
- 1,343
6
votes
3
answers
2k
views
Explicit examples of resolution of (projective) 3-folds over k?
I'm looking for examples of explicit resolutions of (projective) 3-folds over a field k (char 0), with isolated singularities, or at least with smooth singular locus. I've looked in various books and ...
- 183
10
votes
2
answers
834
views
Analytical formula for topological degree
At the first page of the following article http://arxiv.org/pdf/1004.1018v1.pdf [edit: the formula on the arXiv differs from the formula in the published paper, and the formula displayed below is the ...
- 101
17
votes
4
answers
2k
views
Comparing fundamental groups of a complex orbifolds and their resolutions.
Let $X$ be a complex manifold with quotient singularities, and let $\tilde X$ be its resolution (that exists, for example, by Hironaka). Then I am pretty sure that $\pi_1(X)\cong \pi_1(\tilde X)$.
...
- 28.9k
4
votes
3
answers
533
views
Obstructions to being a hyperplane section or a fibre of a Lefschetz pencil
Given a smooth projective variety $X$, when could $X$ fail to be a hyperplane section in some other variety $Y$, or fail to be the fibre of some Lefschetz pencil $\widetilde{Y} \rightarrow \mathbb{P}^{...
- 1,329
12
votes
2
answers
2k
views
Implicit function theorem at a singular point?
Let $F:\mathbb{R}^2 \rightarrow \mathbb{R}$ be three times continuously differentiable in some open neighborhood $\mathcal{U}$ of $(0,0)$. Suppose that $F(0,0) = F_x(0,0) = F_y(0,0) = F_{xy}(0,0) = 0$ ...
- 123
22
votes
5
answers
3k
views
Is a 'generic' variety nonsingular? Or singular?
I'd like to know whether there's some coherent meaning of 'generic' for which one can say that a 'generic' variety over an algebraically closed field $K$, say, is nonsingular or singular. We could ...
- 2,168
3
votes
2
answers
297
views
Singularities of a central fibre of a flat family of smooth surfaces
Suppose I have a one parameter flat family of complex surfaces (regular, of general type) whose general fibre is smooth. Is it possible for the central fibre to have singularities which are not ...
- 63