Questions tagged [singularity-theory]
Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.
554 questions
0
votes
0
answers
82
views
Bijective restriction of the normalization morphism
Let $X$ be an integral separated scheme of finite type over $\mathbb{C}$. Consider the normalization morphism $f:X'\rightarrow X$. Can we always find an affine open $U\subset X'$ such that $f|_U:U\...
0
votes
0
answers
225
views
Generating function with essential singularities
I was recently introduced to analytic combinatorics, and found the method of removing poles astonishing. More precisely, I was reading the last chapter of the popular "generatingfunctionology", in ...
- 5,230
6
votes
0
answers
376
views
Topological Singularities in Affine Varieties
Let $X$ be an affine variety over $\mathbb{C}$. Let $x\in X$.
If $x$ is non-singular, then $x$ is locally holomorphic (in the Euclidean topology). See here for a relevant MO post.
By results of ...
- 8,529
5
votes
0
answers
83
views
Cohomology of the space of generic immersion maps of surface into 3-space
In "Local Invariants of Mappings of Oriented Surfaces Into 3-Space", V.Goryunov classified singular maps of surface into 3-space and considered their resolution and local invariants.
It is natural ...
- 171
2
votes
1
answer
132
views
A special oscillatory orbit in space
Edit: According to the comment of Prof. Eremenko I revise the question.
19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact ...
- 366
3
votes
0
answers
168
views
Is there an affine embedding X for every normal singularity, so that Pic(X)=0?
More formal: Let a normal algebraic singularity be given by a local ring $R$ of finite type. Is there always an affine variety $X$ with a point $x$, so that
$\widehat{\mathcal{O}}_{X,x} \cong \...
- 161
1
vote
1
answer
223
views
Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities?
Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\...
- 2,126
1
vote
1
answer
383
views
Construction of log canonical singularity
I know there's classification about normal log canonical surface singularity in the sense of configuration of exceptional curves.
There is one type of log canonical singularity(not klt) whose ...
- 623
4
votes
0
answers
169
views
Can the rank of harmonic maps decrease far from the boundary?
Let $\mathbb D^n$ be the closed unit ball in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be ...
- 6,741
1
vote
0
answers
93
views
partially simultaneous resolution of singularity
Let X be a projective manifold and let $D=\sum_{i=1}^{m}a_iD_i\in |mL|$ be an effective divisor on $X$ with SNC support. Let $f:X\to Y$ be a surjective morphism over a projective manifold $Y$. Write $...
- 527
4
votes
0
answers
116
views
Bertini-type theorem for strict transform
Let $(X,o)$ be an isolated, normal singularity of dimension at least $3$. Let $\pi: \widetilde{X} \to X$ be a resolution of singularity of $X$. Is it true that for a general hypersurface $H \subset X$ ...
- 2,323
3
votes
0
answers
451
views
Singularities of rational quartic surfaces
Let $X\subset \mathbb{P}^3$ be an irreducible quartic surface, defined over an algebraically closed field $k$. Suppose that $X$ is rational (i.e. birational to $\mathbb{P}^2$). Is is true that $X$ has ...
- 7,722
1
vote
1
answer
93
views
Existence of meromorphic 2-forms over normal surface singularities
Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^...
- 2,032
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
3
votes
0
answers
206
views
An upper bound for the number of singularities of a transversal vector field isometric to the zero field
Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$.
A smooth vector field $X:M \to TM$ is called a transversal vector field if $X(M)$ is transverse ...
- 366
2
votes
0
answers
76
views
Equations needed to define a normal complex surface singularity
This questions is highly related with this other question of mine: Irreducible surface singularity that is not a local set-theoretical complete intersection I just thought that a different look at the ...
- 1,409
7
votes
1
answer
553
views
Relationship between Hilbert-Samuel multiplicity and polar multiplicity
Let $f \in \mathbb{C}[[x,y]]$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $e_f$ of $f$ is given as follows:
$$e_f = \lim_{s \to \infty}\frac{1}{s} \cdot \...
2
votes
0
answers
361
views
Irreducible surface singularity that is not a local set-theoretical complete intersection
I have been looking for a criterion for the germ of an irreducible complex surface singularity $(X,x)$ to be a set-theoretical complete intersection.
A germ $(X,x)$ of an isolated complex singularity ...
- 1,409
2
votes
0
answers
34
views
Smoothings over a real interval
I am asking if somebody knows if the following kind of objects are studied somewhere or if there is some kind of obvious obstruction for them to appear.
Let $(X,0) \subset \mathbb{C}^n$ be a germ of ...
- 83
2
votes
1
answer
89
views
Smoothings of isolated non-irreducible surface singularities
Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing.
Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not ...
- 83
1
vote
0
answers
40
views
On Remmerts reduction
Let $(X,0)$ be a normal surface singularity. An let $\pi: \tilde{X} \to X$ be the minimal resolution. Now, we can apply a result of Oliveira (exploiting previous work by Laufer) and obtain a 1-...
- 1,409
1
vote
1
answer
182
views
Slow and fast forming singularities of the mean curvature flow
Let $M \times [0, T) \to \mathbb{R}^{n+1}$ be a mean curvature flow and let $T$ be a singular time. Let $A$ denote the second fundamental form.
We have a type I singularity if
$$
\max_{p \in M} |A(p,...
- 823
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
1
vote
1
answer
143
views
The surface singularity $x^4=yz$
Sorry if this question is a bit broad. I would like to have examples of papers which have studied the surface singularity
$$x^4=yz,\quad(x,y,z\in\mathbb{C}).$$
I am trying to get a feel about what is ...
- 13
1
vote
2
answers
222
views
Can we extend a logarithmic form to some appropriate compactification?
Given some simple normal crossings divisor $D$ on a complex manifold $X$, which is not assumed to be compact. Given a form $\omega\in H^0(X,\Omega_X^1(\log D))$, when is it possible to find a ...
- 2,788
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
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
1
vote
0
answers
66
views
Deformation sublevel sets of functions which preserve boundary
I'm interested in proving the following fact, which seems to naturally arise from gradient flow deformations, but appears to be a bit tricky.
Consider a smooth family
$$f_s : M \to \mathbb{R}, \quad ...
- 1,227
0
votes
0
answers
123
views
Generalized Sard's lemma
Let $f: X \to \mathbb{R}$ be a $C^{1,1}$ (that is $C^1$ with Lipschitz differential) function on a manifold $X$. Suppose that $f$ is smooth at all points of a subset $C \subset \text{Crit}f$ of ...
- 1,227
6
votes
2
answers
985
views
A paradox on the deformation of singularities
Setup: $\pi: \mathcal X \to C$ is a flat morphism from a germ of a smooth curve $(C, o)$. Suppose the special fiber $\pi^{-1}(o) := \mathcal X_o$ has a certain class of singularities, one wants to ...
- 3,472
2
votes
0
answers
180
views
Pushforward of structure sheaf on quotient surface singularity
Let $f:\mathbb{C}^2 \to \mathbb{C}^2/G$ be a quotient surface singularity. What properties should $G$ have such that the pushforward $f_*\mathcal{O}_{\mathbb{C}^2}$, of the structure sheaf $\mathcal{O}...
- 1,593
1
vote
0
answers
219
views
Desingularization of the zero section of $TM$ as the manifold of singularities of the geodesic flow
However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities"...
- 366
4
votes
0
answers
168
views
Can nonflat deformations of singularities always produce Cohen-Macaulay rings?
To make the question in the title precise, let me phrase it like this. Consider a complete local ring
$$ A := \mathbb{C}[[x_1, \dotsc, x_n]]/(f_1, \dotsc, f_m) $$
and, for definiteness, assume that $...
- 2,663
2
votes
0
answers
78
views
Singularities of quotient of a vector bundle by a lattice
Let $V$ be a complex vector bundle of rank $g$ over an open unit disc $\Delta$ and $H$ be a integral local system of rank $2g$ over $\Delta$ i.e., for every $t \in \Delta$, $H_t \cong \mathbb{Z}^{2g}$....
- 1,593
4
votes
1
answer
257
views
Enumerating the unfoldings of real singularities
Let $f: \mathbb{R}^n \to \mathbb{R}$ be a germ of an isolated, real analytic singularity. Let $I$ be the ideal in $R = \mathbb{R}[x_1, \dots, x_n]$ generated by the components of $\nabla f$, and $Q = ...
- 111
3
votes
1
answer
145
views
Connections between eigenvectors after matrix multiplication
Suppose we have an M$\times$N complex matrix $H$ and its singular value decomposition $H=U\Lambda V^*$ and an N$\times$N covariance matrix $R_s$ with its eigendecomposition $R_s = U_s\Lambda_sU_s^*$. ...
- 33
2
votes
0
answers
674
views
Small contractions as blow ups
To improve my chances of getting answers/ comments I post my mathstackexchange question https://math.stackexchange.com/q/2808852/42548 also here.
I am trying to learn a bit about birational morphisms:...
- 121
3
votes
1
answer
257
views
Bertini-type theorem for reducible schemes
Let $X \subset \mathbb{P}^n$ be a reducible, projective subscheme. Assume that $X$ is reduced (meaning that every local ring is reduced i.e., does not contain nilpotent element). Denote by $S_d$ the ...
- 2,032
2
votes
1
answer
449
views
Example of maps between a smooth curve and a singular curve
I would like an example of maps between a smooth curve $C$ and a singular curve $B $, $f:C \rightarrow B$, where the genus $p_a(C)=p_a(B)$ and greater than or equal to 2.
7
votes
0
answers
506
views
A general definition of an equisingular family of singular varieties?
This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions.
Let $X$ be a ...
- 7,799
1
vote
0
answers
250
views
On regularity of flat families over a DVR
Let $k$ be an algebraically closed field of characteristic zero and $R$ a discrete valuation ring over $k$. Let $\pi:X \to \mathrm{Spec}(R)$ be a flat, projective morphism such that the generic fiber ...
- 1,981
3
votes
1
answer
157
views
Normal form of functions $(x^2+y^2)^n+$ higher terms
By Morse lemma for any $C^{\infty}$ function $f$ on $\mathbb R^2$ with Taylor series $(0,0)$ starting with $x^2+y^2$ one can find local $C^{\infty}$ coordinates $(x',y')$ such that locally $f(x',y')=...
- 14.3k
4
votes
1
answer
321
views
Singularities of the union of two smooth curves
I am looking for a reference that describes what are the possible singularities of a curve $C=C_1 \cup C_2$ which is the union of two smooth irreducible curves $C_1,\,C_2$ on a smooth surface.
Doing ...
- 486
2
votes
0
answers
50
views
Typical singularities of a map from $\mathbb{T}^{m}$ to $\mathbb{R}^{n}$
My question is as follows. Is there known general results on typical singularities (critical points) for smooth maps from $m$-dimensional torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\mathbb{Z}^{m}$ to $\...
- 798
11
votes
2
answers
1k
views
Classification of singularities of plane curves of fixed degree (reference request)
We know the answers to some questions like What is the maximal number of singularities of (reduced) plane curves of degree $d$? for general $d$ (in this case $\tfrac{1}{2}d(d-1)$, obtained by $d$ ...
- 1,124
3
votes
0
answers
146
views
Frobenius structure for A_n singularities
I need to compute monodromy matrices $M(v)$, associated to a Frobenius structure for $A_n$ singularity with flat coordinates $v_1,\dots,v_n$, that is, $f(x)=x^{n+1}$. (due to Saito, Dubrovin etc.) ...
- 31
2
votes
0
answers
149
views
Reference for certain resolution of singularities formulation
I want to use the following resolution of singularity statment as found in Soule et al, Lectures on Arakelov Geometry, p. 40:
$Y$ is a separated algebraic variety of finite type over $\mathbb{C}$, $Z$...
- 195
2
votes
1
answer
646
views
How singular is the metric on an orbifold
I am reading some stuff on orbifolds. I am particularly interested in the metrics on orbifolds. The famous example of one orbifold is the "American football", which is $\mathbb{S}^2$ quotient by the ...
- 711
1
vote
1
answer
295
views
A question about explicit computations of discrepancies
The following is an explicit computation of discrepancies appeared in the book "Birational Geometry of Algebraic Varieties" (Page 126-127) in order to show certain type singularities are not Du Val. ...
- 3,472
12
votes
0
answers
340
views
Homology of Gersten complex for singular schemes
It is one of the important facts in K-theory/motivic cohomology that the Gersten-type complexes (for Quillen K-theory, Milnor K-theory or more generally Rost's cycle modules) are exact for smooth ...
- 17.4k