# Questions tagged [singularity-theory]

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

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### Moduli space of curves away from singular subsets

Recently, I'm interested in the moduli spaces of curves in a (possibly noncompact) complex orbifold (resp. symplectic and almost complex) away from singularities. More specifically, I'm interested in ...
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### Examples of small resolutions in dimension 4 and higher

I have seen numerous examples of 3-folds admitting small resolutions. Are there similar examples in higher dimension? In particular, I am looking for examples of singular varieties of dimension $4$ ...
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### Theories of manifolds w/ extra structure and singularities

Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
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### Any "motive"(-like) theory which can catch that cusp $y^2=x^3$ (and similar) are non-trivial?

Consider cusp $y^2=x^3$ which can also be described as $k[z]~without~z$ , taking $x=z^3,y=z^2$. Algebraically its $Spec$ is quite different from $k$. For example: it has plenty non-trivial "line-...
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1 vote
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### Solutions to ODE/SDE with singular coefficients $dX_t = -X_t/t \, dt + g\,dW_t$

I encountered a question regarding the solutions to SDEs with singular drifts. I searched the literature but had a hard time figuring out the intuition behind these analytic results assuming different ...
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### Solvability of derivation Lie algebras of local finite-dimensional commutative algebras

Let $A$ be a finite-dimensional local commutative algebra (with one) over a characteristic zero field $k$. Is it true that the Lie algebra $\operatorname{Der}_k(A)$ of $k$-derivations of $A$ is ...
• 17.7k
1 vote
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### Contracting a family of rational curves in a Calabi Yau threefold

Suppose we have a Calabi-Yau 3-fold $X$ (not necessarily compact, over $\mathbb{C}$) that contains a ruled surface over a smooth curve $C$ of genus $g$. I am using a strong definition of a ruled ...
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### Are singularities of complex varieties captured by topology?

Let $X \subseteq \mathbb{C}^n$ be an affine complex algebraic variety, with a singularity at some point $x.$ Let $U \subseteq \mathbb{C}^n$ be an open set containing $x$. Can we determine if $x$ is a ...
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1 vote
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### Poisson summation for solutions of the Burgers equation in the form 1/x

Long story short: I'm looking for a good way of showing that the Fourier transform of $1/x$ is a sign function. Motivation and why this has been a problem: I'm dealing with an equation similar to the ...
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### Proof of rationality of roots of Bernstein polynomial à la Lê

Lê Dũng Tráng has a paper "The Geometry of the Monodromy Theorem" (MR0541020 https://mathscinet.ams.org/mathscinet/article?mr=541020 for reference). It's a nice paper, and in it Lê gives a ...
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### Normal forms of ADE singularities

Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms? Does a similar ...
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### Derived category and resolution of singularities

Let $(X,x)$ be an isolated, Gorenstein singularity of dimension at least $2$ and $f: Y \to X$ be a resolution of singularities. Let $E_1, E_2$ be two distinct irreducible components of the exceptional ...
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1 vote
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### What are algebroid curves/branches and their value semigroup?

In “The moduli problem for plane branches”, by O. Zariski, the author defines a plane branch as an irreducible element $f \in \mathbb C[[x,y]]$. In the more recent article "The semigroup of a ...
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### Geometric interpretations of $A_k$ singularities on plane curves

Definition: A real smooth analytic function $f$ defined on some neighborhood of $t_0$ is said to have an $A_k$ singularity at $t_0$ if and only if $$f'(t_0) = f''(t_0) = \dots = f^{(k)}(t_0) = 0$$ ...
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### Modality of a point under a Lie group action

Let $X$ be a smooth manifold and $G$ a Lie group acting on it. V. I. Arnold defines the modality of a point $x\in X$ as follows [1] (see also [2]): We say that a point $x$ has modality $m$ (under the ...
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### How to determine the singlarity type (up to local analytic isomorphism) of a hypersurface surface singularity

Given a polynomial f(x,y,z), it defines a hypersurface in $\mathbb C^3$. I guess there is a classification of hypersurface singularity like Arnold normal form. I wonder given an explicite example of f,...
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### Apparent singularities and non Fuchsian regular points

I am considering the following function of $z$ on the Riemann sphere: $$J(z) = \int_\Delta (L_0+z L_1)^a D^b d^nx$$ where $\Delta \in H_n\big(\Bbb{CP}^n\setminus\{L(x)=0\},S\big)$, $S$ being the ...
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### Deformation to normal cone of the exception divisor of a log-resolution

I am reading the paper Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink due to G. Guibert, F. Loeser, and M. Merle. The main tool, like a lot of papers in ...
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### Definition of canonical pair

Let $(X,D)$ be a pair and $f:Y\rightarrow X$ a log resolution. Write $$K_Y + \widetilde{D} = f^{*}(K_X) + \sum_{i}a_iE_i$$ where $\widetilde{D}$ is the strict transform of $D$. I found the following ...
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1 vote
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### Induced resolution of singularities

I am not a specialist in singularity theory but currently I have to touch resolution of singularities and I'd like to know whether I have understood Hironaka's theorem correctly. Let $k$ be a field of ...
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1 vote
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### Converse of transfer theorem : does asymptotic behaviour of coefficients describe the singularity?

I'm interested in Singularity Analysis after reading this Math.Stack post. In the "Analytic Combinatorics" book by P. Flajolet and R. Sedgewick, at p.390, the following transfer theorem (...
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### Curvature explosion and metric landmark stability

$\newcommand{\prin}{\mathrm{prin}}$Context: Let $S$ be the unit sphere in some finite dimensional vector space $V$. Given a connected compact Lie group real representation $\rho:G\rightarrow O(V)$, ...
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