Questions tagged [singularity-theory]

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

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Hironaka's proof of resolution of singularities in positive characteristics

Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier... Unlike Atiyah's paper, Hironaka's paper does not have a ...
Henry.L's user avatar
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Microlocal geometry - A theorem of Verdier

(1) In "Geometrie Microlocale", Verdier states the following theorem. Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$. Then for $\ell$ a linear form on $E$, we have a ...
AFK's user avatar
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12 votes
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Homology of Gersten complex for singular schemes

It is one of the important facts in K-theory/motivic cohomology that the Gersten-type complexes (for Quillen K-theory, Milnor K-theory or more generally Rost's cycle modules) are exact for smooth ...
Matthias Wendt's user avatar
12 votes
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570 views

Cohomology and conifold transition for the quintic

Let $Y\subset \mathbb{C}P^4$ be the quintic threefold given by the equation $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4+5X_0X_1X_2X_3X_4=0$$ it has 125 singular points whose links are homeomorphic to $S^2\times S^...
David C's user avatar
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11 votes
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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 (...
Pulcinella's user avatar
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10 votes
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385 views

Can I compute the cohomology of the complement of a log canonical divisor as if it were normal crossings?

Let $X$ be a smooth projective variety and $D$ a log-canonical divisor and let $U = X \setminus D$. I have heard the slogan "log-canonical is just as good as normal crossings for Hodge theory". This ...
David E Speyer's user avatar
9 votes
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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 ...
ormula's user avatar
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550 views

Singularities arising from the Minimal Model Program (an algebraic point of view)

I will start the story by the end: Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ?...
Pedro Montero's user avatar
9 votes
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Jacobian ideals reference

Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ...
Karl Schwede's user avatar
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A criterion for rational singularities in mixed characteristic

Let $R$ be a mixed characteristic discrete valuation ring with perfect residue field and $f:X \to \mathrm{Spec}(R)$ a flat proper morphism. If the generic fibre of $f$ is smooth and the special fibre ...
naf's user avatar
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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 ...
Katharina Hübner's user avatar
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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}...
MCS's user avatar
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Gysin exact sequence for a singular subvariety

Let $k$ be an algebraically closed field (I'm interested in a characteristic $p>0$ specific example) and let $X$ be a (smooth if needed) algebraic variety. Let $Y \subset X$ be a (possibly) ...
Oblomov's user avatar
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Which presentations of (non)planar algebras give rise to knots?

Reidermeister's theorem states that the set of knots, modulo ambient isotopy, is isomorphic to the planar algebra generated by crossings, modulo Reidemeister moves. This planar algebra presentation is ...
Daniel Moskovich's user avatar
7 votes
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A general definition of an equisingular family of singular varieties?

This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions. Let $X$ be a ...
Saal Hardali's user avatar
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Higher homotopy of diffeomorphism groups from singularities

In the case of a genus $g$ surface $\Sigma$, it is well known that $MCG(\Sigma) = \pi_0 \operatorname{Diff}^+(\Sigma)$ is generated by Dehn twists, which come from a Kahler degeneration with smooth ...
Nati's user avatar
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242 views

Singularities of an analytic function over a non-archimedean field

What do we know about the types of singularities that a convergent power series over a non-archimedean field can have? More specifically: i) What types of essential singularities can occur? ii) Are ...
Helene Sigloch's user avatar
7 votes
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154 views

When is a smooth function locally equivalent to a truncation of its Taylor series?

Let $U \subset \mathbb{R}^n$ be an open set and let $f:U \rightarrow \mathbb{R}$ be a smooth proper function. For $p \in \mathbb{R}^n$ let $$T_p(x_1,\ldots,x_n) = \sum_{i_1,\ldots,i_n=0}^{\infty} c_{...
Lisa's user avatar
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Isolated singularities and tangent cones

Assume that I have an affine hypersurface $X =V(f)\subset \mathbb{C}^4$ of degree $d$ with an isolated singularity of multiplicity $m$ at the origin $o=(0,0,0,0)$. Let $$f:=f_m + f_{m+1}+ \cdots +f_d$$...
Francesco Polizzi's user avatar
7 votes
0 answers
528 views

Physicists Euler number conjecture

Physicist's Euler number conjecture says: If $G \subset SL(n,\mathbb{C})$ is a finite group, $X=\mathbb{C}^n/G$ is the quotient space and $f:Y \rightarrow X$ a crepant resolution (always exists for $...
Mohammad Farajzadeh-Tehrani's user avatar
7 votes
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516 views

An elementary question in singularities

The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ...
Daniel Pomerleano's user avatar
6 votes
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189 views

Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$

In my research I encountered automorphisms of the ring of convergent power series $$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$ which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
red_trumpet's user avatar
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forms on singular spaces that can be integrated on an LCI

I'd like to characterize rational $k$-forms on a singular scheme $X$ which can be integrated on any (real) bounded submanifold that is locally LCI in $X$ in a real sense (i.e., which is the real ...
Dmitry Vaintrob's user avatar
6 votes
0 answers
206 views

Resolution graph of higher dimensional ADE singularities

I am looking for different configurations of the exceptional divisors arising from blowing up a higher dimensional ADE singularity (see p. 240 of this article of Bruns for a description of such ...
user43198's user avatar
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6 votes
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362 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'\...
AG learner's user avatar
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6 votes
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352 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 ...
Sean Lawton's user avatar
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A topological property of flat morphisms

Let $f\colon X\to Y$ be a faithfully flat morphism of smooth projective varieties over $\mathbb{C}$. Assume that the generic fiber of $f$ is smooth. Then there exists a non-empty Zariski open subset $...
asv's user avatar
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6 votes
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226 views

Toric Degenerations and Nearby Cycles

Suppose that $f: X \to \mathbb{A}^1$ is a toric degeneration in the sense of Nishinou-Siebert. In other words let X be a (possibly singular) toric variety equipped with a (not necessarily proper) ...
Justin Hilburn's user avatar
5 votes
0 answers
198 views

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 ...
Will Sawin's user avatar
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5 votes
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206 views

Is the Taylor map continuous?

(Skip to the bolded theorem below for my question, if you'd like) Some context on asymptotic expansions and the Taylor map In the setting of irregular singularities of meromorphic connections on the ...
Brian Hepler's user avatar
5 votes
0 answers
232 views

Is there a way to solve this integral on the sphere explicitly?

Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that $k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral $$f(y):=\int_{\...
Medo's user avatar
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0 answers
203 views

Stability of nodal hypersurfaces

We denote by $\Pi_{n,d}$ the space of homogeneous polynomials of degree $d$ in $n+1$ variables $x_0,\ldots,x_n$, i.e. $\Pi_{n,d}=\Gamma(\mathbb{P}^n(\mathbb{C}),\mathcal{O}(d))$. The group $G=SL(n+1)$ ...
Nikolay Konovalov's user avatar
5 votes
0 answers
160 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 ...
Giulio's user avatar
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5 votes
0 answers
399 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 ...
xin fu's user avatar
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0 answers
140 views

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 ...
Matthias Wendt's user avatar
5 votes
0 answers
83 views

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 ...
this_is_a_banana's user avatar
5 votes
0 answers
293 views

Resolving analytic normal crossings singularities

Let $X$ be a non-singular (complex) variety and $Y \subset X$ be a (reduced) irreducible subvariety with only normal crossings singularity (locally, in the analytic topology, the singularity is ...
user43198's user avatar
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5 votes
0 answers
668 views

Ehresmann's without properness in the algebraic category?

Ehresmann's theorem for manifolds states: If $f : X \to Y$ is a proper submersion, then $X$ is a locally trivial fibration on $Y$. Some sources I am reading (Lazarsfeld Positivity in Algebraic ...
Elle Najt's user avatar
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5 votes
0 answers
174 views

Framed singular knots

I've recently run across what one might (and I suspect people probably do) call framed singular knots, or maybe singular ribbon knots. Regardless of the name, what I mean is the following: Let $D$ be ...
Paul's user avatar
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5 votes
0 answers
184 views

Real structure in the mixed Hodge structure associated to an isolated singularity

We know that a mixed Hodge structure on a complex vector space $H$ with an integral lattice $H_{\mathbb Z}$ consists of the weight filtration and the Hodge filtration. For an isolated hypersurface ...
Guangbo Xu's user avatar
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5 votes
0 answers
333 views

Stratification of a smooth map

So, this is an exercise. But from math.stackexchange I have been suggested to post this question here. To find the Thom-Boardman stratification of the smooth map $f(x,y,a,b,c,d)=x^2y+y^3+a(x^2+y^2)+...
PepeToro's user avatar
  • 231
5 votes
0 answers
334 views

Fixed point sets that carry topology

Let $M$ be a closed smooth manifold. A generic diffeomorphism $\phi: M\rightarrow M$ has non-degenerate fixed points, i.e. the intersections of its graph with the diagonal in $M\times M$ are all ...
Andre Carneiro's user avatar
4 votes
0 answers
252 views

Vanishing cycles and injectivity of the specialisation map

Consider a proper algebraic map between complex varieties $f : X \to D$ ($D$ is the unit disk), which is a submersion over $D^*$. I would like to know if they are any condition on $f$ such that the ...
Nicolas Hemelsoet's user avatar
4 votes
0 answers
79 views

Possible number of zeros of a stable perturbation of a germ $(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$

Let $f:(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$ be an analytic germ. Assume that it has isolated zero at 0, that is, $f^{-1}(0)=\{0\}$, what is more, assume that the dimension of the local algebra* $Q(...
Pintér Gergő's user avatar
4 votes
0 answers
156 views

How to judge whether an orbifold is good

My own case comes from dynamic system on compact complex manifolds. To be precise, let $M$ be a compact complex 3-dimentional manifold, $W^c$ a holomorphic foliation of M with 1-dimentional uniformly ...
Chicken feed's user avatar
4 votes
0 answers
316 views

Relationship between exact symplectomorphisms and Hamiltonian diffeomorphisms (in the context of Milnor fibers)

Here is a preamble/setup. Suppose we have a symplectic manifold $(M,d\lambda)$ with an exact symplectic form. By Stokes theorem, $M$ must have nonempty boundary. An exact symplectomorphism $\phi:M \to ...
inkievoyd's user avatar
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4 votes
0 answers
72 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 $\lambda \in \mathbb C^\mu$...
Nick Salter's user avatar
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4 votes
0 answers
105 views

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{...
User43029's user avatar
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4 votes
0 answers
163 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 ...
Asaf Shachar's user avatar
  • 6,611
4 votes
0 answers
114 views

Bertini-type theorem for strict transform

Let $(X,o)$ be an isolated, normal singularity of dimension at least $3$. Let $\pi: \widetilde{X} \to X$ be a resolution of singularity of $X$. Is it true that for a general hypersurface $H \subset X$ ...
user45397's user avatar
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