# Questions tagged [singularity-theory]

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

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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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 $$\... 0answers 50 views ### Robust features intuition? The terminology robust features was introduced by Ian Porteous as they are features of a surface wich be followed when the surface is deformed. They capture important aspects of the surface geometry. ... 0answers 104 views ### On resolution of singularities over an Artin ring For a locally noetherian scheme X, Grothendieck conjectured that if X is quasi-excellent then there is a proper birational map Y \to X s.t. Y is regular. We now fix an Artin ring R whose ... 1answer 298 views ### Computations of divisor class monoids Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors". Let A be a (commutative) domain, K its field of fractions. A ... 2answers 393 views ### restricting the “Whitney” map \newcommand\R{\mathbb R}Suppose f:\R^2 \to \R^2 is a Whitney map with singularities (well, I'm not sure if this is the name for it, Whitney calls them excellent maps in his 1955 paper), i.e. it is ... 0answers 157 views ### Blowing up the zero section for “Chasse au Canard” (some new kind of geometric canards) In this paper "Canard cycles and center manifolds" one encounters the blowing up of a non isolated set or manifold of singularities of a vector field or a singular foliation. This is a ... 1answer 154 views ### Is it possible that \int |f^2(z)|^{t+1}(P\bar{P})\phi(z) dz=0 for all compactly supported \phi? When proving that the log-canonical threshold is minus the largest root of the Bernstein-Sato polynomial one considers the integral$$\int |f^2(z)|^{t+1}(P\bar{P}(\phi(z))) dz,$$where P is a linear ... 2answers 187 views ### Milnor lattice and Du Val singularity I am reading this paper: https://arxiv.org/abs/0810.2687 by A. J. de Jong and Robert Friedman. In the proof of Theorem 4.10, a singularity of the following type shows up$$y^2=x^3+z^{6d-1}.When d=... 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 ... 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 ... 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 ... 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\... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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, ... 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 \... 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 (... 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 ... 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 ... 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 ... 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 ... 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 ... 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 \... 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 ... 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 \... 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} ... 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 "... 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... 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-... 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 ... 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'\... 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 ... 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),$$where f_i : (\... 0answers 125 views ### Reference request: Singular curves I'm interested in coherent sheaves on a singular curve.(For example, global dimension, Serre duality, Riemann-Roch's theorem for singular curves,etc....) I find treatment of it only in Hartshorn's ... 1answer 223 views ### Riemann-Hurwitz for real maps Let S be a (compact, connected) Riemann surface of genus g and f: S\to \mathbb CP^1 be a degree d meromorphic function. Then the Riemann-Hurwitz formula tells us that the number of ... 1answer 156 views ### Singular locus of a linear system of hyperplane sections Let X\subset\mathbb{P}^N be a rational smooth projective irreducible non degenerated variety of dimension n=\dim(X) and let$$\mathcal{H}=|\mathcal{O}_X(1) \otimes\mathcal{I}_{{p_1}^2,\dots,{p_l}^...
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 ...