Questions tagged [singularity-theory]

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

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101 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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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 "Elementary ...
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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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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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1answer
63 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 $...
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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'\...
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Applying analytic coordinate changes to singular function germs

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 ...
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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 : (\...
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109 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 ...
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1answer
208 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 ...
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1answer
110 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}^...
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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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1answer
79 views

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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143 views

canonical divisor on singular curves with nodal point

What's the definition of canonical divisor(or whatever related concept) on singular curve with nodal point. More generally, what the definition of canonical divisor on a singular variety X, which is ...
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1answer
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Non-example for Whitney (a) stratifications

Given a $C^1$ stratification $\mathscr{S}$ of a $C^1$ manifold $M$, we write $N^\ast \mathscr{S}$ for the union of conormals to the strata. The stratification is said to be Whitney (a) if $N^\ast \...
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When is the coarse moduli space of genus $g$ stable curves smooth?

Let $\overline{\mathcal{M}}_g$ be the moduli stack of genus $g$ stable (nodal) curves and let $\overline{M}_g$ denote its coarse moduli space. In 1969, in the paper "The irreducibility of the space of ...
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151 views

Can we approximate a vector field on the plane with non-vanishing vector fields in $W^{1,2}$?

Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated. Does there exist a sequence of ...
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321 views

Derived Category of the derived critical locus, is it the category of Matrix Factorizations?

Let $W \in \mathbb{C}[x_1, \dots, x_n]=R$ be a polynomial with an isolated critical point at the origin. A Matrix Factorizations for $W$ consists a $\mathbb{Z}/2\mathbb{Z}$-graded finite free $R$-...
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Scheme-theoretic image and delta-invariants

Let $(X,o)$ be an affine, isolated, normal, Gorenstein singularity. Let $f$ and $g$ be two morphisms from $\mbox{Spec}(\mathbb{C}[[t]])$ to $X$ (also known as formal arcs) such that the closed point ...
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1answer
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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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832 views

Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have high rank?

Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth. Suppose that $df$ is invertible outside a set of Hausdorff dimension $\le n-1$, and ...
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565 views

Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have distinct singular values?

$\newcommand{\SO}[1]{\text{SO}(#1)}$ $\newcommand{\dist}{\operatorname{dist}}$ Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth. Set $$...
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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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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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68 views

Log Kodaira dimension of Briekorn varities

Is there any formula or estimate of the log-Kodaira dimension of the Brieskorn variety $V_{a_0,\ldots,a_n}:=\{x_0^{a_0}+\ldots + x_n^{a_n}=1\}$ for $2\le a_0\le \ldots \le a_n$. In particular, I ...
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1answer
327 views

Essential singularity [closed]

In shaum's outline complex analysis,definition of essential point is: An isolated singularity that is not pole or removable singularity is called essential singularity Now in the same book there is an ...
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414 views

Surprising appearances of Painlevé transcendents

What are some of your favorite examples of enumerative problems whose answer ended up being (related to) a solution to one of the Painlevé equations? I have seen examples from enumeration of classes ...
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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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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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1answer
187 views

Kähler-Einstein metrics on singular varieties

Let $X$ be a normal projective variety with klt singularities with numerically trivial canonical divisor $K_X$. Does there always exist a Kähler-Einstein metric on $X$?
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Link of a singularity

I would like to understand the topological type of a link of a singularity in a simple example. Consider for instance the cone ${xy-z^2=0}\subset\mathbb{C}^3$. If we set $x = x_1+ix_2, y = y_1+iy_2, z ...
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51 views

Removability of the isolated singularity of real analytic mappings with nondegenerate Jacobian

Let $B^n=\{x\in \mathbf{R}^n: |x|<1\}$$(n>2)$. Consider real analytic mappings $f_1:B^n\setminus \{0\}\to B^n$, $f_2:B^n\setminus \{0\}\to \mathbf{R}^n$ and $f_3: B^n\setminus \{0\}\to S^n$ ...
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1answer
213 views

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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On the milnor number of analytic germ map

If $f:(\mathbb{C}^n,0) \to (\mathbb{C},0)$ is an analytic germ with an isolated singularity, then the Milnor number of $f$, denoted by $\mu(f)$ can be defined as $\dim_{\mathbb{C}} \mathcal{O}_n/\text{...
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Singularities of curves that are moving

Let $k$ be an algebraically closed field, let $d\ge 2$ be an integer and let $f,g\in k[x,y,z]$ be two homogeneous polynomials of degree $d$ without common factor. We want to know what are the ...
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309 views

Bertini type theorem for very ample line bundle

Let $X$ be a normal, projective variety (can take $X$ to be a hypersurface in a projective space) of dimension at least $3$. Let $L$ be a very ample line bundle on $X$, hence base-point free. What can ...
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How can I find the integral orthogonal group of a given symmetric positive definite form?

I wonder how one can study the integral orthogonal group of a given (symmetric, positive definite) bilinear form like the one described by the following matrix: $$M=\begin{bmatrix} x_1 &...
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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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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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118 views

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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54 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\...
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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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275 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 ...
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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 ...
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111 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 ...
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152 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 \...
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1answer
139 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 $s \in S$, $\...
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1answer
175 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 ...
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156 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 ...
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72 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 $...

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