Questions tagged [singularity-theory]

Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.

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3
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1answer
117 views

A convolution type singular integral operator with log

Define a convolution type operator $T_m$ by $$T_m(f) = p.v.\int_\mathbb{R}f(x-y)\frac{\log^m|y|}{y}dy.$$ Here $m\ge0$ is an integer. Consider $f \in H^s (s > 0)$ which is the usual Sobolev space. ...
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1answer
169 views

Is there a “minimal” Whitney stratification of a complex hypersurface?

Let $X\subset \mathbb C^n$ be a complex hypersurface (given by $F=0$ where $F$ is a polynomial). It is known then that $X$ admits a Whitney stratification. This is a decomposition of $X$ into smooth ...
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0answers
48 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 ...
5
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1answer
165 views

Computing the invariants of ball quotient surfaces

The two-dimensional complex unit ball $B$ has group of biholomorphic automorphisms $PU(2,1)$. If $Γ$ is an arithmetic subgroup of $PU(2,1)$, the quotient $Γ\text{\\}B$ is an orbifold. Taking its ...
2
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1answer
290 views

When is a birational bijection étale?

So this question is probably not "research level", although, for what it is worth, it is coming up in a research paper I am presently writing. Let $X,Y$ be irreducible affine varieties over $...
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0answers
18 views

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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134 views

Resolution graph of higher dimensional ADE singularities

I am looking for different configurations of the exceptional divisors arising from blowing up a higher dimensional ADE singularity (see p. 240 of this article of Bruns for a description of such ...
4
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1answer
203 views

Higher direct image for rational singularities

Let $X$ be a normal, projective (complex) variety with at worst rational singularities. Let $\pi:Y \to X$ be the resolution of singularities obtained by blowing-up the singular points. Is $R^1 \pi_*\...
2
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1answer
107 views

Bertini type result for complete intersection varieties containg a non-singular variety

Let $X$ be a smooth, projective $\mathbb{C}$-variety of dimension $r$ containing in $\mathbb{P}^n$. Denote by $I_X \subset \mathbb{C}[X_0,...,X_n]$ the ideal of $X$ defined by some homogeneous ...
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0answers
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Maslov cycle for the Conley-Zehnder index - what are its regular points?

I'm looking at the definition of the Conley-Zehnder index, where it is important to look at the group $$\text{Sp}(2n)^* := \{ A \in \text{Sp}(2n) | \det (A-\text{Id}) \neq 0 \}$$and its complement $$\...
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0answers
50 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. ...
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104 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 ...
5
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1answer
298 views

Computations of divisor class monoids

Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors". Let $A$ be a (commutative) domain, $K$ its field of fractions. A ...
4
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2answers
393 views

restricting the “Whitney” map

$\newcommand\R{\mathbb R}$Suppose $f:\R^2 \to \R^2$ is a Whitney map with singularities (well, I'm not sure if this is the name for it, Whitney calls them excellent maps in his 1955 paper), i.e. it is ...
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157 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 ...
3
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1answer
154 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 ...
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2answers
187 views

Milnor lattice and Du Val singularity

I am reading this paper: https://arxiv.org/abs/0810.2687 by A. J. de Jong and Robert Friedman. In the proof of Theorem 4.10, a singularity of the following type shows up $$y^2=x^3+z^{6d-1}.$$ When $d=...
9
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0answers
114 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 ...
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1answer
155 views

Non-transverse intersection of submanifolds

What can we tell about non-transverse intersection points of (smooth) submanifolds? Especially, in the case of complementary dimensions, how can we define and calculate the ''multiplicity'' of an ...
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1answer
91 views

Numerical methods for evaluating singular integrals

The Helmholtz decomposition for a vector field B contains both volume integrals and two boundary integrals (https://en.wikipedia.org/wiki/Helmholtz_decomposition). For brevity I show just one of the ...
5
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1answer
104 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\...
4
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1answer
246 views

singular metric (with essential singularity)

Working on some $Q$-curvature equation in dimension $4$, I have been faced with singular metric of the form $(\mathbb{B}, e^{-1/\vert x\vert ^2} \vert dx\vert)$. I try to figure out to what those ...
5
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0answers
86 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 ...
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0answers
17 views

lifting of control data along a stratified morphism

Let $f:X\to Y$ be a stratified map between Whitney stratified spaces such that for each stratum $S$ of $Y$, $f:f^{-1}(S)\to S$ is a proper stratified submersion. Let $\mathscr{T}_Y$ be a Thom-Mather ...
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0answers
19 views

Stratification which makes the defining functions isotrivial

Let $0\in X\subset\mathbb{C}^N$ be a germ of complex space and $0\in Z\subset X$ be a closed analytic subset (globally) defined by holomorphic functions $f_1,\dots,f_r$. Is there a complex analytic ...
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0answers
144 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 ...
3
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1answer
125 views

Holomorphic vector fields with a non-degenerate isolated zero

Let $v$ be a holomorphic vector field defined in a neighbourhood of $0$ on $\mathbb C^n$ with an isolated zero at $0$. Let $\sum_{i,j}{a_{ij}}z_i\frac{\partial}{\partial z_j}$ be the linear term of $...
7
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1answer
374 views

J.-P. Serre: Duality of regular differentials on singular curves

I already asked this on math.stackexchange.com, but didn't get any responses. I hope it is appropriate here. Let $X'$ be an irreducible singular algebraic curve over an algebraically closed field $k$, ...
2
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0answers
224 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 $\...
11
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0answers
305 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 (...
7
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0answers
183 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 ...
7
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1answer
329 views

General conditions for normality of blow-up

Let $X$ be an integral, affine, normal complex surface. I am looking for conditions on zero-dimensional closed subschemes $Z$ in $X$ such that the reduced scheme associated to the blow-up of $X$ along ...
5
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0answers
118 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 ...
10
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1answer
258 views

Fixed point scheme of finite group Cohen-Macaulay?

Let $X$ be a quasi-projective scheme over a field $k$. Let $G$ be a finite group acting on $X$ whose order is invertible in $k$. If $X$ is Cohen-Macaulay, can we conclude that the subscheme of fixed ...
2
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0answers
206 views

Definition of Q gorenstein variety

I have a question about the definition of Q-Gorenstein variety. I saw a definition of Q-Gorenstein variety:for a normal variety $X$, it's Q-Gorenstein if the canonical divisor is Q Cartier. I wonder ...
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0answers
157 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 $\...
3
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1answer
307 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 ...
5
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1answer
155 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 \...
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0answers
154 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 $X^{\vee}$ ...
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0answers
111 views

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 "...
4
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0answers
51 views

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 $\lambda \in \mathbb C^\mu$...
2
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0answers
79 views

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-...
1
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1answer
83 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 $...
6
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0answers
244 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
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1answer
95 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 ...
7
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1answer
206 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 : (\...
2
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0answers
125 views

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 ...
6
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1answer
223 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 ...
1
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1answer
156 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}^...
2
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0answers
53 views

Approximate a one-form with nowhere vanishing one-forms having bounded Laplacians

This is a follow-up question of this one. Let $\mathbb{D}^2$ be the closed two-dimensional unit disk (endowed with some smooth Riemannian metric), and let $\sigma \in \Omega^1(\mathbb{D}^2)$ be a ...

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