Questions tagged [singularity-theory]
Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.
554 questions
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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 ...
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2
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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 ...
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Can harmonic maps with immersive boundary conditions have singular points?
Let $\mathbb D^2$ be the closed unit disk in $\mathbb R^2$. Let $f:\mathbb D^2 \to \mathbb{R}^2$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^2 \to \mathbb{R}^2$ be ...
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singularities of the dual variety of a surface
I am looking for a proof/reference of the following simple fact, which I think it holds true.
Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the ...
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1
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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 \...
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0
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On Whitney's paper on real algebraic varieties
I had previously asked this question on math.stackexchange and did not receive an answer and so I decided to reword it and pose it here.
This question is based on Whitney's paper titled "...
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2
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1
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Family of curve singularities whose generic members are smooth
Let $f: (X,x)\rightarrow (\mathbb C,0)$ be a deformation of a curve singularity $(X_0,x)$, and let $f: X \rightarrow T$ be a sufficiently small representative. Assume that $(X,x)$ is reduced and pure ...
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Poll about your proof of resolution of singularities and a request for advice
The questions first: What is the proof of resolution of singularities that you know?
Why am I asking?: There are a number of proofs of resolution of singularities of varieties over a field of ...
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1
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How to measure how much a rational function/a singularity of variety is complicated?
There are some theorems about various zeta functions which states the rationality of those.
For example, when you consider Igusa's zeta function, roughly the generating series of solutions mod $p^n$ ...
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2
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construct the elliptic fibration of elliptic k3 surface
Hi all,
As we know, every elliptic k3 surface admits an elliptic fibration over $P^1$, but generally how do we construct this fibration? For example, how to get such a fibration for Fermat quartic?
...
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2
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Which weighted projective spaces (and their finite quotients) are local complete intersections?
Let $G$ be a finite subgroup of $\textrm{Gl}_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of $\mathbb{...
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Approximate a one-form on the disk with nowhere vanishing one-forms satisfying an asymptotic vanishing of some derivatives
Let $\mathbb{D}^2$ be the closed two-dimensional unit disk, and let $g:\mathbb{D}^2 \to \mathbb{R}$ be a non-constant harmonic function (smooth up to the boundary).
Does there exist a sequence of ...
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"Arithmetic genus" of a plane curve singularity.
I believe that the following questions are very basic, but I don't know how to get a reference.
Consider a curve in the plane $C\in \mathbb C^2$ with a singularity at $0$ and suppose it is
...
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Reference request: Singular curves
I'm interested in coherent sheaves on a singular curve.(For example, global dimension, Serre duality, Riemann-Roch's theorem for singular curves,etc....)
I find treatment of it only in Hartshorn's ...
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Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?
Let $X$ be an affine variety. Let $Y$ be smooth and let the map $f\colon Y\rightarrow X$ be proper birational. We will call $Y$ a smooth resolution of $X$.
Do the cohomology groups $H^i(Y,\mathcal{O}...
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Stratified pseudomanifold
In the definition of an $n$-dimensional stratified pseudomanifold one demands the following filtration
$X=X_n \supset X_{n-1}=X_{n-2} \supset X_{n-3}\supset ... \supset X_0 \supset X_{-1}=\emptyset$. ...
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1
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Intersection multiplicity via parametrization in general
My question is a generalization of Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide.
Take two complex space germs $(A, 0)=V(I_A)$ of dimension $...
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Giant Rat of Sumatra singularity
I would be grateful for explanations of the issues raised in any
of these three questions, or pointers to the relevant literature
(now updated with answers):
How did a particular singularity come to ...
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2
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0
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About the regularity of Thom's first isotopy theorem
Consider an abstract stratified set $(V, \Sigma)$ in the sense of Thom-Mather
(see Mather's note page 491-492 https://www.ams.org/journals/bull/2012-49-04/S0273-0979-2012-01383-6/S0273-0979-2012-01383-...
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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')=...
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Elementary short exact sequence of sheaves
This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow A^2/\...
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Question of terminology concerning singularities of transversal type A2
If we consider a plane curve that is a Legendrian front with a singularity of type A2, we say that this singularity is a cusp. If we consider a surface of (for instance) R^3 that is a Legendrian front ...
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Canonical scheme structure on the singular locus of a variety
I first asked this question on Math StackExchange but no answers were given.
Let $X$ be subvariety of affine space $\mathbb{A}_{k}^n$, where $k$ is a field, and suppose $X$ is given by equations
$$...
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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 ...
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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 ...
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1
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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 $s \in S$, $\...
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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 ...
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minimal resolution of singularities
What is the minimal resolution of singularities of the surface
$S^2(X^3+Y^3+Z^3)-3(S^2+T^2)XYZ=0$ which is a subset of $\mathbb{P}^1\times\mathbb{P}^2$
Please note that in this equation $[S:T]\in{\...
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1
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About 3-fold log canonical singularity
As far as I know, log canonical surface singularities were classified. How about higher dimensional case?
I especially want to know whether given 3-fold singularity is log canonical or not.
Let $f$ ...
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0
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Asymptotics of a certain integral in singularity theory
Let $f:\mathbb{C}^2\to \mathbb{C}$ be an isolated plane curve singularity. Consider the versal deformation space $\mathbb{C}^\mu$ parameterizing deformations $f_\lambda$ for $\lambda \in \mathbb C^\mu$...
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Ordinary n-uple Points and Resolution of Singularities on a Surface
Let $X$ be an algebraic variety over some algebraically closed field $\Bbbk$ and let us assume $\dim(X)=2$, i.e. $X$ is an algebraic surface.
First, I would like to know the definition of an ordinary ...
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Milnor Number of real and imaginary parts of holomorphic germs?
By performing some computations using the Singular software, I've noticed the following pattern: if $\mu$ is the Milnor Number of a holomorphic germ $f\in \mathcal{O}_n$ at the origin, then the Milnor ...
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6
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Advantage of discrepancy
In the definition of Minimal model of projective variety, some authors use of discrepancy, and some others omit this condition. I am wondering to know the advantage of discrepancy In the definition of ...
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vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves
I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:
$H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
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1
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Can you prove Givental's conjecture on wavefronts and the icosahedron?
In his remarkable book The Theory of Singularities and its Applications, Vladimir Arnol'd discussed a conjecture of A. B. Givental, which asserts that the symmetry group of the icosahedron is secretly ...
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What formal properties should resolution of singularities have?
If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...
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0
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Local complete intersection on a smooth variety
Let $X$ be a smooth variety and let $x_{1},\ldots,x_{k}$ be general points. Let $T:=<T_{x_{1}}X,\ldots,T_{x_{k}}X>$ be the join of the tangent spaces at the points $x_{1},\ldots,x_{k}$. If we ...
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Normalization of affine curves in singular surfaces
Let $X$ be a normal, isolated surface singularity with $x_0 \in X$ the unique singularity. Let $C \subset X$ be a hyperplane section i.e., defined by a single equation. Denote by $n:\widetilde{C} \to ...
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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 ...
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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 ...
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Padé Approximants of Power Series with Natural Boundaries
Consider a power series $\sum_{n=0}^{\infty}c_{n}z^{n}$ for which $c_{n}\in\left\{ 0,1\right\}$ for all $n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}...
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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 ...
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2
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Log canonical counterexample to Kawamata-Viehweg vanishing
I found in the literature that, in characteristic 0, Kodaira vanishing holds for log-canonical pairs. On the other hand, the usual statement for Kawamata-Viehweg vanishing talks about a klt pair $(X,\...
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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 ...
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Failure of devissage vs link topology in algebraic K-theory
This is somehow related to (or maybe a simplified version of) an earlier question (see here) regarding Gersten complexes for singular varieties. The Gersten complexes arise from the coniveau spectral ...
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2
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1
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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
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Holomorphic vector fields tangent to a hypersuface singularity
Let $(V,0)\subset (\mathbb{C}^n,0)$, $n\geq 3$ a (germ of) hypersurface given by $V =\{f=0\}$, $f$ a germ of holomorphic function. A (germ of) holomorphic vector field $X$ on $(\mathbb{C}^n,0)$ is "...
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0
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Is the normalized derivative of a holomorphic function Sobolev?
This question is a cross-post from MSE. it is also a special case of this question.
Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$, and let $f:B \to \mathbb{C}$ be holomorphic on the interior $B^o$, and ...
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4
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1
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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 = ...
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1
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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 ...
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