Questions tagged [singularity-theory]
This tag is for questions relating to Singularity Theory. In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes.
374 questions
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Codimension of singular points of hypersurface $Y\subset X$
Let $X$ be a complex manifold, and let $Y\subset X$ be an irreducible hypersurface (i.e. analytic subset of codimension 1). Lemma 2.3.22 in Huybrechts' Complex Geometry shows the sheaves $\mathcal{O}(-...
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Can an immersion from the disk to a surface "double up" on its boundary?
I would like to find a simple proof or counterexample to the following claim, which has come up in some work we are doing related to curves in surfaces which bound immersed disks.
Let $F$ be any ...
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Singular locus of Noetherian normal domain of Krull dimension 2
Let $R$ be a Noetherian normal domain of Krull dimension $2$. By Serre's criteria, if $\mathfrak p$ is a prime ideal such that $R_{\mathfrak p}$ is not regular, then $\mathfrak p$ must have height $2$,...
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Value of algebraic generating function at $z=1$
We are given $p(z,u)$, a nonzero polynomial with real coefficients. Suppose we know that: (a) there is a generating function $g(z)$ that solves $p(z,g(z))=0$; (b) $g(z)$ has nonnegative coefficients; (...
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Slicing pseudomanifolds and getting sub-pseudomanifolds?
Smooth manifolds like $\mathbb{R}^n$ and $S^n$ behave nicely under slicing by coordinate hyperplanes — for example, slicing $\mathbb{R}^3$ along $x_3 = c$ yields $\mathbb{R}^2$, which is again smooth. ...
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Is the singular support of a distribution always measure zero?
An example of a distribution whose singular support has measure zero is the delta function $δ(x)$, whose singular support is just $\{0\}$. I don't know of any examples of distributions whose singular ...
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Puiseux series for solution of polynomial equations
I want to study the behavior of a curve $\mathcal{C}$ implicitly defined as the zero of function $\mathbf{F}: \mathbb{R}^n\rightarrow\mathbb{R}^{(n-1)}$, $\mathcal{C}:\{\mathbf{x}\in \mathbb{R}^n \...
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Is the the preimage of a cusp in a projective rational curve always a singleton?
I'm working with projective rational curves over an algebraically closed field $k$ (for simplicity, $k = \mathbb C$. More specifically, they are projective curves $C\subseteq \mathbb P^n$ with a ...
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Asymptotic for the roots of a Polynomial
I am studying an eigenvalue problem on an Hilbert space. I turn it into a first order dynamical system. I need the asymptotic behavior of that asymptotic system. To do that, I am faced with computing ...
- 1,282
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Is every compact analytic subset of $\mathbb{C}^n$ finite?
"Let $\mathbb{C}^n$ be adopted with the Euclidean topology and $A$ be and analytic subset of $\mathbb{C}^n$ (which means that for any $z\in \mathbb{C}^n$ there are an open neighborhood of $z$ in $...
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On the blow-up of a cone at the origin
I'm trying to solve an excercise on the blow-up of the affine cone $\mathcal{Z}(xy-z^2)\subset \mathbb{A}^3$ at the origin.
Let $\sigma:X'\rightarrow \mathbb{A^3}$ be the blow-up of the affine space ...
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Equations for surface quotient singularities
Recall that a surface quotient singularity is given by $(\mathbb{C}^2/G,0)$ where $G$ is a small finite subgroup of $U(2)$. If $G$ is a subgroup of $SU(2)$, then the singularity is actually du Val ...
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Dimension of Jacobian Algebra (in Singularity Theory)
Motivation: In Singularity Theory, the Milnor Number is defined to be the dimension of the Jacobian algebra and is finite in the case of an isolated singularity.
We define the Jacobian algebra of a ...
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Do we have general polynomial equation $f(x,y) = 0$ of degree $d$ which has cusp singular point at origin?
Do we have general polynomial equation $f(x,y) = 0$ of degree $d$ which has cusp singular point at origin?
I want to explicitly compute something in singularity theory for the cusp $X = Z(f) \subset \...
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Two definitions of a normal complex surface
I am looking for the definition of a "normal surface singularity". I could not find such a definition even in the book with title Normal Surface Singularities (https://link.springer.com/book/...
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