# Questions tagged [singularity-theory]

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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 \$...