Questions tagged [singularity-theory]
Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.
554 questions
5
votes
1
answer
263
views
Measuring contact between algebraic varieties
I have two regular surfaces in three space, both of which are given by an equation. I would like to measure the contact between the two surfaces using only their equations. Usually, one would find a ...
- 126
1
vote
0
answers
105
views
How would you call a variety that is locally a complete intersection up to defect c?
Let $X$ be an equidimensional variety of dimension $n$ over a field that can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in $P^N$ (for a large enough $N$; we ...
- 16.9k
3
votes
1
answer
236
views
Counting nodal singularities on a surface
How many lines in $\mathbf{P}^5$ passing through a fixed point $p$ meet in at least two points a fixed smooth surface $S$ given by the intersection of three quadrics?
Or equivalently, calling $T$ the ...
- 453
3
votes
0
answers
265
views
Good covers on complex algebraic varieties with normal crossings singularities
Let $X$ be a topological space. A good cover on $X$ is an open cover such that all finite non-empty intersections are contractible. It is a theorem of Hironaka that (complex) algebraic sets admit ...
- 879
2
votes
1
answer
355
views
T^i functors are isomorphic for analytically isomorphic isolated singular points
I've been having trouble proving the following:
Let $B$ and $B'$ be local rings, essentially of finite type over $k$, having isolated singularities at the closed points. Suppose that they are ...
- 2,108
2
votes
1
answer
583
views
Brieskorn's proof of a theorem by Milnor about the Milnor number
I am looking for a reference or short explanation of a proof by E. Brieskorn.
In his famous work "Singularities of complex hypersurfaces" Milnor proves that the (nowadays called) Milnor Number (in ...
- 1,124
2
votes
0
answers
491
views
Explicit Computations of Dynkin Diagrams of Isolated Singularities
Let $f$ be a complex polynomial with an isolated singularity at the origin. Take a Morse deformation $\tilde{f}$, and consider the braid group action on the set of distinguished bases of vanishing ...
0
votes
0
answers
101
views
on lifting elements in a tangent space
Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$
Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and $...
- 3,472
1
vote
1
answer
128
views
Finite construction of lacunary functions using algebraic and certain analytic operations
Algebraic functions have a discrete set of singularities. Lacunary functions, e.g. $f(z)=\sum_{n=0}^\infty z^{2^n}$, have a continuum of singularities at every point of the boundary of their disk of ...
- 13
2
votes
2
answers
285
views
How can I compute the full set of nodes of a surface?
The nodes of a surface are special cases of more general singularities. For example, the Cayley cubic has four nodes.
The full set of singularities of a surface can be characterized by finding all ...
2
votes
0
answers
98
views
Is it obvious that the defining conditions to obtain a particular singularity are well-defined on the quotient space?
Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function
vanishing at the origin, with
the following properties:
$$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 \...
- 3,245
1
vote
1
answer
194
views
Adding singular equations to a smoothing of a hypersurface singularity
Let $f \in \mathbb{C}[x,y,z]$ be a polynomial which defines an isolated singularity $0 \in D:= (f=0) \subset \mathbb{C}^3$.
Assume that $\mathcal{D}:= (f+tx =0) \subset \mathbb{C}^3 \times \mathbb{C}$...
- 909
1
vote
1
answer
174
views
Is the space of degree $d$ curves with marked smooth points dense inside the space of curves with marked points?
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d} $ be the space of nonzero
homogeneous degree $d$ polynomials in three variables upto scaling, where
$\delta_d = \frac{d(d+3)}{2} $
(basically degree $...
- 3,245
2
votes
1
answer
233
views
Is there an invariant similar to the delta invariant that distinguishes an $A_2$ node form an $A_1$ node?
Consider the following question: If two nodes collide what do you get?
First of all it can not be a strict $A_2$ node, because the delta invariant
of that is $1$. So it has to be more singular than ...
- 3,245
3
votes
1
answer
760
views
Corank 4 hypersurface singularities
A function f: ($\mathbb{C}^n$,0) $\to$ ($\mathbb{C}$,0) is considered a hypersurface singularity if the point $(0,0,\dots,0)$ is the only point in the ideal $\langle \frac{\partial f}{\partial x_1}, \...
- 508
0
votes
1
answer
617
views
What is the simplest way to show that a section of a vector bundle is transverse to the zero set
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$, be the space of homogeneous
degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where
$\delta_d = \frac{d(d+3)}{2}$. ...
- 3,245
7
votes
0
answers
535
views
Physicists Euler number conjecture
Physicist's Euler number conjecture says:
If $G \subset SL(n,\mathbb{C})$ is a finite group, $X=\mathbb{C}^n/G$ is the quotient space and $f:Y \rightarrow X$ a crepant resolution (always exists for $...
3
votes
0
answers
215
views
cotangent complex for finite flat morphism and different ideal
Let a ring $A$, $F=A[X_{1},\dots X_{n}]$ and $B:= F/J$.
We suppose that we have a finite flat lci morphism $f:Spec(B)\rightarrow Spec(A)$.
To mesure the singularities of this map, Gabber-Ramero ...
- 3,472
2
votes
1
answer
346
views
Does the Newton polytope characterize the equisingular i.e topological type?
Whenever, people talk about singular plane curves they talk about their Newton polytope which is obviously coordinate dependent. I understand that with some conditions over the singular curve, some ...
- 115
4
votes
0
answers
443
views
What is known about "singularity types" in the Murphy's Law sense?
In his "Murphy's Law" paper, Vakil gives a definition equivalent to the following:
The singularity type of a pointed scheme $(X,p)$ its equivalence class, under the following equivalence relation: $...
- 7,338
2
votes
0
answers
91
views
Complexity of mappings (forms) in R. Thom's "Structural stability and morphogenesis"
In his "Structural stability and morphogenesis", R. Thom (especially in the chapter about dynamics of forms) among other things speculates about a notion of complexity of a "form" (mapping between ...
- 192
4
votes
1
answer
827
views
When singular points of a reduced scheme are not dense in it?
A stupid AG question: could singular (Zarisky) points be dense in a reduced (Noetherian) scheme $S$? If yes, which 'standard' restrictions on $S$ could ensure that this does not happen? For example, ...
- 16.9k
2
votes
1
answer
397
views
Is resolution of singularities effective?
Suppose I have a singular projective variety defined by some homogeneous equations in complex projective space. Is the resolution of singularities effective? That is, do I actually know which smooth ...
user2529
2
votes
1
answer
431
views
Stable singularities of smooth map $\mathbb R^3\to \mathbb R^4$
Does anybody know any classification of stable singularities of smooth map $f:\mathbb R^3\to \mathbb R^4$?
It is clear that there are singularities which look like intersection of 2 (or 3 or 4) ...
- 5,063
0
votes
0
answers
61
views
Disturbing regular level submanifold of a smooth function
Let $a$ be a regular value of a smooth function on a closed manifold and $\{f=a\}$ a corresponding level submanifold. It is known that any such function can be approximated by a Morse function $g$. ...
- 75
3
votes
1
answer
229
views
What is the simplest way to represent a $D_5$ singularity?
Consider this curve $f(x,y)=0$ given by
$$ f(x,y) := y^3 + y^2 x + x^4 =0.$$
Is it obvious that after a change of coordinates near the origin, this
curve is equivalent to
$$ \hat{y}^2 \hat{x} + \...
- 3,245
2
votes
0
answers
103
views
Lagrangean equations for the generating function of quadrangulations
Let $M(z)$ be the generating function of edge-rooted connected quadrangulations, with $z$ marking the number of edges. I derived the following Lagrangean equations for $M(z)$:
$$M(z) = \psi(L(z)),~\...
- 21
4
votes
2
answers
627
views
The link of a singular quintic hypersurface in CP^4
Given a family of quintic hypersurfaces in $\mathbb{CP}^4$ by
$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5+(5+\epsilon)x_1x_2x_3x_4x_5$
we get a singular variety for $\epsilon=0$ with 125 singular points.
I know ...
- 93
1
vote
1
answer
244
views
How can one bound 'the lower perverse degree' for a constant sheaf on a variety that is smooth in high codimension?
Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of ...
- 16.9k
1
vote
1
answer
500
views
Does any one understand the details of M Kazarian's work in enumerative geometry of $\mathbb{C}\mathbb{P}^2$ ?
I wanted to know if anyone understood the details of the paper
"Multisingularities, cobordisms, and enumerative geometry" available at the site
http://www.mi.ras.ru/~kazarian/.
In particular does ...
- 3,245
5
votes
0
answers
343
views
Fixed point sets that carry topology
Let $M$ be a closed smooth manifold. A generic diffeomorphism $\phi: M\rightarrow M$ has non-degenerate fixed points, i.e. the intersections of its graph with the diagonal in $M\times M$ are all ...
1
vote
1
answer
103
views
On solution of a recursion with rectangular matrices
Greetings to members here.
The question is how to calculate the solution $S(k)$ of the following recursive equation
$$J(k)S(k+1)J^{T}(k)=A(k)S(k)A^{T}(k)+R(k)$$
where $J$ and $A$ are rectangular not ...
- 97
1
vote
0
answers
154
views
How do I check whether an orbifold admits deformations?
(Cross-post from math.stackexchange, where it has received no attention.)
Orbifolds $\mathbb{C}^2/\mathbb{Z}_n$, given by the action $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$ with $\zeta$ a primitive $...
- 884
1
vote
0
answers
88
views
Three- dimensional astroid and catastrophe maps
Is a three-dimensional astroid curve $(2\cos^3u,2\sin^3u,3\cos2u)$ a part of a bifurcation set of some catastrophe map?
- 11
0
votes
0
answers
112
views
Homology basis of minimal resolution
Let $S$ be a compact algebraic surface with du Val singularities and $T$ be the minimal resolution of it. Is it true that
$$
H_2(T,\mathbb{Z})/(\oplus_{i}\mathbb{Z}E_i)\cong H_2(S,\mathbb{Z}),
$$
...
1
vote
1
answer
599
views
complex singularity exponent, lct
Hi everybody,
I have a question about log canonical thresholds / complex singularity exponents.
If I understood well, this invariant sees more things than the multiplicity, for example, the cusp in ...
1
vote
0
answers
183
views
Smoothing of a hyperquotient singularity
Let $f$ be a polynomial in $k$ complex variables, and suppose the affine variety $V$ given by $f = 0$ has an isolated singularity at the origin, but is otherwise smooth. Now assume that some cyclic ...
- 884
7
votes
0
answers
518
views
An elementary question in singularities
The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ...
- 3,758
3
votes
1
answer
419
views
Is there an upper bound and a lower bound on the contribution to the genus, for a singularity of codimension k?
To make my question precise, suppose you have a complex curve locally given by
$$f(x,y) =0 $$
and $f$ has singularity of type $\chi_k$ at the origin. The codimension of
this singularity is $k$. Let ...
- 3,245
2
votes
0
answers
102
views
semicontinuity of the conductor defined by Temkin
We say a principal pair $(X,\mathcal{I})$ where $X=Spec(A)$ is affine scheme and $\mathcal{I}=\tilde{I}$ where $I\subset A$ is a principal ideal generated by $\pi$ wich is a non zero divisor.
For a ...
- 3,472
4
votes
1
answer
240
views
Transversals to singular subvarieties
Say $\mathbb{C}^d \subset Y^{N-k} \subset \mathbb{C}^N$ are closed imbeddings of complex analytic subvarieties of the indicated dimension, $Y$ is not smooth. At a point $y \in Y$, a generic, ...
- 8,723
3
votes
0
answers
185
views
Equivalence of Level Sets
Consider the zero level set of $f : \mathbb{R}^3 \to \mathbb{R}$, where $0$ is a regular value. Consider also the space of planes passing through the origin, i.e. $\mathbb{RP}^2$. For a fixed plane $P ...
- 126
3
votes
0
answers
90
views
on Neron defect of smoothness for groups schemes
Let $G$ a semisimple simply connected group over $\mathbb{C}$.
Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$.
We consider $I_{\gamma}$ the group ...
- 3,472
4
votes
1
answer
336
views
How to enumerate curves with a singularity once you know the corresponding Thom Polynomial?
Does any one know how to go from "Thom polynomials" to "Enumerating curves". I believe there
is a relation between the two, but I don't know how to go about it. Let me make the
question more concrete....
- 3,245
4
votes
1
answer
618
views
Singularity theory references
I am looking for some good references on singularity theory. I'm interested in singularity theory in the context of mirror symmetry, so this means I'm interested in things like Picard-Lefschetz theory,...
0
votes
1
answer
487
views
Log resolutions of linear series
Let $X$ be a complex normal projective variety, let $|L|$ be a non empty linear series on $X$ and let $b(|L|)$ be its base ideal.
Suppose $f:X'\rightarrow X$ is a log resolution of the ideal $b(|L|)$.
...
- 1,150
2
votes
0
answers
271
views
When does the smoothing of projectivized tangent cone lift to a deformation of a space?
Let $(X,0)\subset(\mathbb{C}^N,0)$ be the (formal) germ of a singular space (isolated singularity). Let $\mathbb{P}T_{(X,0)}\subset\mathbb{P}^{N-1}$ be its projectivized tangent cone (considered as a ...
- 2,232
0
votes
1
answer
87
views
Question regarding closure of sets defined by the vanishing of holomorphic functions
Consider the following subsets of $\mathbb{C}^n$ given by
$$ X := \{x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 \} $$
$$ Y := \{ x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 \} $$
where $f, g$...
- 3,245
2
votes
0
answers
174
views
Are there ways to make low degree checks for enumerative formulas except for curves in CP^2?
This is a concrete question in Enumerative geometry. Let $S$ be a compact
complex surface and $L\rightarrow S$ a holomorphic line bundle. Let
$$ \delta_d = \text{dim}~ \mathbb{P}(H^0(S,L^d)) $$
...
- 3,245
3
votes
0
answers
677
views
Sebastiani-Thom isomorphism for D-modules
Considering $f:X\to \mathbb{C}$, $g:X\to \mathbb{C}$ and $f\oplus g:(x,y)\mapsto f(x)+g(y)$.
The Sebastiani-Thom isomorphism is an isomorphism $\Phi_{f\oplus g}(M\boxtimes N) = \Phi_{f}(M) \otimes \...
- 7,527