Questions tagged [singularity-theory]
Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.
554 questions
4
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Action of the monodromy on the cycle made of the real points
Let $f : \Bbb C^n \to \Bbb C$ be a polynomial function with real coefficients.
Let $X_t = f^{-1}(t)$ denote the fiber above some $t \in \Bbb C$. Let assume that the set of real points of $X_t$, for $t ...
- 1,044
3
votes
0
answers
82
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Singularities of fibrations 2
This question is related to my previous question:
Singularities of fibrations
Assume that $X$ is a complete intersection irreducible $3$-fold in a product of projective spaces. So that $X$ is ...
- 8,998
4
votes
3
answers
775
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A necessary and sufficient condition for a curve to have an $A_k$ singularity.
Hi
Does any one know of a necessary and sufficient condition for a curve to have a singularity
of type A_k.
More precisely, a curve f=0 has a singularity of type A_k at a point, if there exist
local ...
- 3,245
3
votes
1
answer
477
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Simple example of isolated critical point with non-semisimple monodromy
Consider a polynomial map $f :\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ with $f(0)=0$ (no constant term) and with isolated critical point at $0 \in \mathbb{C}^{n+1}$. We can choose a disc $D$ of some ...
user42325
3
votes
1
answer
346
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Analytically but not algebraically smoothable singularity
Are there examples of algebraic singularities which may be smoothed analytically but not algebraically? It certainly seems possible, but if not, why? Are there conditions under which this becomes true,...
- 1,493
4
votes
2
answers
611
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Vanishing associated to a resolution of singularities
Let $\pi: V\to W$ be a resolution of singularities, let $E \subset V$ be the exceptional divisor, and let $F$ be a coherent sheaf such that $R^i\pi_*F=0$ for $i>0$.
Can we conclude that $R^i\...
- 2,218
3
votes
0
answers
243
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Does the link of a hypersurface singularity determine its analytic type?
Consider a hypersurface $V(f) \subseteq \mathbb{C}^{n+1}$ with an isolated singularity at the origin. If $L := V(f) \cap S^{2n+1}_\epsilon$ is the link of $V(f)$ (with $S^{2n+1}_\epsilon$ a ...
- 3,058
1
vote
0
answers
68
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Poisson cohomology of germfied Poisson structures in dimension two
Let $f(x, y)$ be a smooth function in the real case or a holomorphic function in the complex case. Denote $\pi=f（x, y)\frac{\partial}{\partial x}\wedge \frac{\partial}{\partial y}$ be the ...
- 57
6
votes
1
answer
1k
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Blowing-up an ordinary double point, then contracting the exceptional locus to a curve
Let $X\subset\mathbb P^4$ a projective hypersurface with an ordinary double point at $o\in X$.
Blow-up $\mathbb P^4$ at $o$ and let $E\simeq\mathbb P^3$ the exceptional divisor of this blow-up. ...
- 7,902
5
votes
1
answer
546
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For what varieties do we have results on the category of singularities?
Let $X$ be a singular variety. Define the (triangulated) category of singularities (as in Orlov's paper)
as the Verdier quotient of the derived category of coherent sheaves on $X$ modulo the full ...
- 1,423
2
votes
0
answers
98
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Singularities of $Spec(Sym^* E^{\vee})$ for $E$ a coherent sheaf
Let $X$ be a smooth complex algebraic variety, and $\mathscr{E}$ a torsion-free coherent sheaf on $X$.
Which type of singularities can the total space $\mathrm{Tot}(\mathscr{E}):=\underline{\mathrm{...
- 23.4k
8
votes
2
answers
573
views
Small neighborhoods of singularities on varieties
In Singular points of complex hypersurfaces, John Milnor proves the following theorem:
Let $x \in V$ be a point on a variety $V$ in $\mathbb{R}^n$ or $\mathbb{C}^n$. Assume $x$ is either a smooth ...
- 7,338
7
votes
3
answers
2k
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What does being Analytically Isomorphic imply for classification of singularities on curves?
Hartshorne I.5 mentions the definition of being analytically isomorphic:
P on X and Q on Y are analytically isomorphic iff the completion of O_P is isomorphic to the completion of O_Q where the ...
- 3,804
6
votes
1
answer
175
views
Is there an algorithm to find out the number of small solutions to a polynomial equation, when we vary all the coefficients?
Let $\Phi (z,t)$ be a polynomial given by
$$ \Phi(z,t) := z^n + A_{n-1}(t) z^{n-1} + \ldots + A_1(t) z + A_0(t).$$
Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation
$$ \...
- 3,245
2
votes
0
answers
186
views
Singularities of algebraic curves, and torsion of the pull-back of the differential module by the normalisation
The problem in the following :
given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field $\...
- 223
1
vote
0
answers
189
views
A definition of arithmetic divisor with conic singularities?
I have a question related to the preprint "Heights and metrics with logarithmic singularities" by G. Freixas i Montplet.
Let $X$ be an arithmetic variety with arithmetic divisor $D$ how can we ...
user21574
7
votes
2
answers
1k
views
Can one prove vanishing of higher direct images fiber-wise?
Let $\pi:X\to Y$ be a proper map of algebraic varieties (over $\mathbb C$) which is a bi-rational equivalence.
are the following statements equivalent?
The derived direct image of $O_X$ is $O_Y$.
...
- 2,649
6
votes
1
answer
267
views
Are codimension one foliations of $\mathbb{R}^{n}-\{0\}$ with compact leaves, stable at origin?
Assume that we have a codimension one foliation of $\mathbb{R}^{n}-\{0\}$ with compact leaves.
Is it true to say that the foliation is stable at origin:That is: for every neighborhood $V$ of $0$,...
- 366
3
votes
0
answers
1k
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Blowdown and contraction
I am sorry, my question is very naive.
2nd Edit: Let us suppose that $V$ is a smooth complex projective variety, and $Y\subset V$ is a smooth divisor and has an ample conormal line bundle.
We would ...
- 9,870
3
votes
2
answers
1k
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Higher dimensional nodes
A node on a curve is a singular point that locally looks like the intersection of two lines. I think the precise way to say this is that $p \in X$ is a (closed?) point on a scheme $X$ (of finite type ...
- 173
5
votes
4
answers
546
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Smooth in codimension-k and the weight filtration
Let $X$ be an algebraic variety. Then $H_{et}^k(X)$ has a filtration whose associated graded pieces are labeled by "weights", certain integers between $0$ and $2k$. If $X$ is smooth, then the weights ...
- 156k
10
votes
0
answers
573
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Singularities arising from the Minimal Model Program (an algebraic point of view)
I will start the story by the end:
Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ?...
- 987
4
votes
0
answers
898
views
A strong form of implicit function theorem (what happens when the derivative is degenerate?)
(this can be considered as some ad)
Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
- 2,232
2
votes
1
answer
235
views
Is there a formula for the number of rational cuspidal curves in surfaces other than P^2?
Let $M$ be a two dimensional compact complex manifold and $A \in H_2(M, \mathbb{Z})$
a fixed homology class. Define a rational curve in $M$ to be $\textit{1-cuspidal}$ if the singularities of the ...
- 3,245
1
vote
0
answers
74
views
Simple question about surface singularities
Given $\epsilon \in (0,1)$, is it possible to find two finite familes $\mathcal{F}$ and $\mathcal{P}$ of weighted graphs, such that the weighted graph of the minimum resolution of any $\epsilon$-klt ...
- 1,595
4
votes
2
answers
560
views
Is there an analogous concept for the degree of a map, when the spaces are singular?
Let $M$ and $N$ be two smooth compact, oriented manifolds and
$X\subset M$ an oriented submanifold of $M$ of dimension $k$
(not necessarily closed). Suppose in addition that $\bar{X}-X$ is contained ...
- 3,245
6
votes
0
answers
233
views
Toric Degenerations and Nearby Cycles
Suppose that $f: X \to \mathbb{A}^1$ is a toric degeneration in the sense of Nishinou-Siebert. In other words let X be a (possibly singular) toric variety equipped with a (not necessarily proper) ...
- 1,543
1
vote
0
answers
581
views
generalization Abhyankar's lemma
This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question.
Let ...
- 7,320
1
vote
1
answer
926
views
triple point singularity
Assume a complex surface $X$ admits a fibration structure over $\mathbb{CP}^1$ with some singular fiberes. Are there explicit examples of such surfaces with triple point singularity?
- 87
0
votes
0
answers
126
views
When is a critical value of a map contained in the interior of the image?
Let $M^n$ be a compact manifold, and $F\colon M \to \mathbb{R}^n$ a smooth map. The inverse function theorem implies that every regular value of $F$ lies in the interior of $F(M)$, hence every point ...
- 321
2
votes
0
answers
114
views
Orbit spaces of Coxeter groups and singularities
I have often seen in the literature the statement that the orbit spaces of irreducible finite Coxeter groups are equivalent to unfoldings of singularities.
For instance, taken from Dubrovin, ...
- 1,821
2
votes
1
answer
360
views
Controlling singularities on log mmp
Suppose all my varieties are complex threefolds $X\rightarrow Y$ over some smooth base curve germ $Y$. We can assume the fibres are Del Pezzo surfaces with generic smooth fibre.
If I do (relative) ...
- 2,215
27
votes
2
answers
1k
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Limit of a series of singularities
The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities $x^...
- 5,846
3
votes
1
answer
567
views
Deformations of quotient singularities
Let $Y$ be an affine scheme over a field of characteristic zero. Suppose we have a group $G$ acting on $Y$ and that the subset of $Y$ of points with non-trivial stabilizer is in codimension greater or ...
- 8,998
3
votes
1
answer
326
views
fearful of defining equivalent germs for non isolated singularities
Two power series $G(x_1, \ldots, x_n)$ and $F(x_1, \ldots, x_n)$ are equivalent over $\mathbb{C}$ if there is an automorphism of the ring
$\mathbb{C}[[x_1, \ldots, x_n]]$ given by $x_1 \to \phi(x_1, \...
3
votes
0
answers
324
views
Implicit function theorem for singularities
I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild".
For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a ...
0
votes
1
answer
286
views
A condition on isolated singularity
Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = 0$...
- 1,207
1
vote
2
answers
1k
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crepant resolution
Let's put $m>n$ two nonnegative integers and $Gr:=Grass(n,k^m)$ the grassmanian of the subspaces of dimension $n$ in $k^m$. We have a natural immersion $Gr \subset P({\Lambda}^{n} k^m)$ and I call $...
- 543
2
votes
2
answers
461
views
Producing $(-2)$ curves on a smooth surface
We know that blowing up a point on a surface produces a $(-1)$ curve. Is there any such standard techniques to produce $(-2)$ curves in a smooth surface?
- 2,032
2
votes
1
answer
2k
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de jong's alteration theorem for families
What is the current status of de Jong's smooth alteration theorem for a family of schemes?
His 1997 paper shows that given any family of curves $X/S$ with $S$ of finite type (and, say, local) over a ...
- 7,972
7
votes
0
answers
246
views
Singularities of an analytic function over a non-archimedean field
What do we know about the types of singularities that a convergent power series over a non-archimedean field can have?
More specifically:
i) What types of essential singularities can occur?
ii) Are ...
- 1,450
1
vote
0
answers
197
views
Is a variety a local complete intersection if it is locally a complement of to a smooth $N$-dimensional affine of $N-m$ affine subvarieties?
If an equidimensional variety $V$ of dimension $m$ is locally a set-theoretic complete intersection (i.e., it can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in $P^...
- 16.9k
3
votes
0
answers
269
views
Hypersurface with singularities
I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?).
It was said ...
- 5,063
2
votes
2
answers
617
views
Bounds for the milnor number of a hypersurface singularity
I am having a hard time in finding an upper bound in terms of the degree and the dimension for the Milnor number of an isolated hypersurface singularity. I am mostly interested in surfaces on the ...
- 943
6
votes
2
answers
646
views
How can we find a surface with a given singularity?
I was surprised the first time I learned that a quintic plane curve can have an $A_{10}$ singularity i.e $x^2+y^{10}$. I am wondering if there is something about that phenomenon: Given a singularity ...
- 943
9
votes
0
answers
2k
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Jacobian ideals reference
Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ...
- 20.5k
2
votes
3
answers
594
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Alterations factor as modification + finite map
I'm learning about de Jong's theory of resolution of singularities and the following fact is used numerous times: an alteration of varieties $h: X \rightarrow Y$ factors as $X \xrightarrow{\pi} Z \...
- 2,108
2
votes
1
answer
710
views
General position argument
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be ...
- 3,245
3
votes
2
answers
322
views
Surfaces in $\mathbb P^3$ with many simple isolated singularities
Could anybody help me with examples of surfaces $X\subset\mathbb P^3$ (projective, over $\mathbb C$) having many isolated singularities of the type $A_1$ ($x^2+y^2+z^2=0$) or $A_2$ ($x^2+y^2+z^3=0$) ...
- 1,939
1
vote
0
answers
542
views
Presence of singular points in the trajectory of a double pendulum
Watching the trajectory of a double pendulum, I caught myself wondering if it would be possible to prove that the path the second pendulum makes contains "cusps" or singular points. Upon investigating ...
- 109