Questions tagged [singularity-theory]
Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.
574 questions
4 votes
0 answers
214 views
An infinite sequence of birational proper morphisms
Let $X$ be a projective normal variety and $X=X_1 \to X_2 \to \ldots$ be an infinite sequence of birational proper morphisms of normal proper varieties. Is it true that $X_i \to X_{i+1}$ is an ...
- 483
0 votes
0 answers
103 views
Multiplicity at a point of a parameterised algebraic variety
Let $f_1,\dots,f_{n},g_1,\dots,g_n\in\mathbb C[t_1,\dots,t_{n-1}]$ polynomials of degree $d$ such that $V(g_1,\dots,g_{n})=\emptyset$. Consider the map $$\begin{array}{cccc} \psi:& \mathbb A^{n-1} ...
- 83
1 vote
0 answers
190 views
Examples of Poincaré spaces
Recall, an algebraic variety $X$ of dimension $n$, is called a Poincaré space if the associated cohomology groups satisfy Poincaré duality. I am looking for examples of Poincaré spaces. I found some ...
- 2,467
3 votes
1 answer
227 views
Existence of Lefschetz pencil in singular varieties
Given a smooth, projective variety $X$, we know that there exists a Lefschetz pencil parameterizing divisors in $X$ such that the fibers are either smooth or has at worst ordinary double point ...
- 2,467
0 votes
0 answers
66 views
Concordance group of stable maps from $S^1$ to $S^1$
It seems that the concordance group of stable maps from $S^1$ to $S^1$ is computed and is given by the topological degree of these maps, that is, is 𝑍. However I don't agree with this statement ...
- 31
4 votes
1 answer
257 views
Rational smoothness and local intersection cohomology
I know that if the Kazhdan-Lusztig polynomial $P_{uv}=1$ for all $u<v$, then the Schubert variety $X_v$ is rationally smooth. If I understand correctly, the $KL$ polynomial can be written in terms ...
- 669
2 votes
0 answers
213 views
What is the "correct" definition of a local deformation?
Short version: If a local deformation of $X$ is any flat morphism $\pi$ into the spectrum of a local ring together with an isomorphism of the special fiber and $X$, there is no bound on how "bad&...
6 votes
0 answers
202 views
Sufficient infinitesimal conditions to ensure that $f \colon \mathbb{R}^n \rightarrow \mathbb{R}^N$ is locally injective at $0$
Let $f \colon \mathbb{R}^n\rightarrow \mathbb{R}^N$ be smooth (assume it is analytic if it helps). I am looking for infinitesimal conditions on $f$ at $0$ (i.e. condition on the derivatives and ...
- 1,329
2 votes
0 answers
149 views
What are all the possible Bernstein-Sato polynomials?
Recall that to any polynomial $f(x_1,\dots,x_n)$ in $n$-variables, one can associate a $1$-variable polynomial $b_f(s)$ called the Berntein-Sato polynomial. It is known that $b_f(-1) = 0$ and moreover ...
- 8,081
5 votes
1 answer
409 views
Intersecting a smooth hypersurface with planes
Let $X\subseteq \mathbb{P}_{\mathbb{C}}^n$ be a smooth hypersurface of degree $d\ge 4$, and consider the Grassmanian $G=\operatorname{Gr}(3,n+1)$ parametrizing the ($2$-)planes inside $\mathbb{P}_{\...
- 211
3 votes
0 answers
84 views
Realise a deformation of a complex singular complete intersection of dimension zero as a real variety
Let $(X,0)$ be the germ of a complete intersection of dimension zero in $(\mathbb{C}^n,0)$. Assume that $(X,0)$ is singular. This is a case of what is called an ICIS (Isolated Complete Intersection ...
- 363
6 votes
2 answers
325 views
For which varieties does the diagonal locally require the minimal number of equations?
Let $V$ be a variety of dimension $d$ over a (say) algebraically closed field $K$. Which assumptions ensure that the diagonal ($\cong V$) in $V\times V$ can be locally set-theoretically ...
- 17.5k
3 votes
0 answers
251 views
Singularities of Algebraic Curves
As far as I know, the singularities of plane curves with degree $\leq7$ can be classified, see Chapter 2 of Makkoto Namba's text book 'Geometry of Projective Algebraic Curves'. Also there are ...
- 635
4 votes
0 answers
102 views
On Whitney regularity of rank stratifications
Let $f:X \to Y$ be a smooth (i.e., $C^\infty$) map between compact smooth manifolds of dimensions $m = \dim X$ and $n = \dim Y$. We can partition $X$ by the rank of $f$'s derivative, i.e., define $$...
- 15.9k
2 votes
0 answers
229 views
Rational singularities vs rationally smooth
Let $X$ be a normal algebraic variety over $\mathbb{C}$. What is the relationship between: -$X$ has rational singularities, i.e. for any resolution of singularities $f:\tilde{X} \to X$, we have $R^*f_*...
- 1,025
3 votes
1 answer
234 views
Tangent space to a subset of a Euclidean space as a space of derivations, and possibility of having an exponential map for manifold with singularities
I'm attempting to define the tangent space at a point to any embedded subset $X$ of a Euclidean space $\mathbb{R}^D, $ Motivation: I'm currently working on a problem with manifold with singularity, ...
- 1,661
1 vote
0 answers
113 views
Singularities of a ramified cover of a smooth variety
Let $\mathbb{A}^m$ denote $m$-dimensional affine space over an algebraically closed field $k$ of char $0$. Consider the action of the symmetric group $S_n$ on $(\mathbb{A}^m)^n$ which acts on $(x_1,......
- 11
1 vote
0 answers
259 views
Are singularities of semistable models nice?
Let $X$ be a semistable curve defined over a number field $K$. Moreover, let $\mathcal X\to S=\operatorname{Spec }O_K$ be a (normal) semistable model of $X$. In this setting I would like to understand ...
- 159
1 vote
0 answers
109 views
Is there embedded resolution up to quotient singularities in characteristic zero?
Definition 7.5 on page 34 here indicates that over a field of characteristic zero, one can perform embedded resolution up to replacing a singular subvariety with one with just simple normal crossings. ...
- 1,106
1 vote
0 answers
107 views
When is $\mathbb{Q}_\ell[d]$ coconnective for the perverse $t$-structure on a projective variety?
Let $X$ be a projective variety of dimension $d$ over a field. If $X$ is smooth then $\mathbb{Q}_\ell[d]$ is a perverse sheaf. Is it true in greater generality that $\mathbb{Q}_\ell[d]$ is in $D_{\le ...
- 1,106
2 votes
0 answers
193 views
Name and properties for the "local number of extra equations" on a variety
I need a number that measures how far a variety $V$ is from being a local set-theoretic complete intersection. Is there any name for numbers of this sort; does any literature treat them or something ...
- 17.5k
1 vote
0 answers
111 views
Pullback of the canonical divisor along blowup of a singular surface and intersection numbers
Given $X$ a normal projective surface over $\mathbb{C}$ and $p$ be a singular point of $X$ . Let $\pi: Y \rightarrow X$ be the blowup of $X$ at the point $p$. Is it true that $\pi^{\star}(K_{X}) \...
- 75
9 votes
1 answer
681 views
Fibers of generic smooth maps between manifolds of equal dimension
I have heard that the following is a "well-known" Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite ...
- 603
1 vote
0 answers
121 views
Projection from a point and singularity
Let $X \subset \mathbb{P}^n$ be a hypersurface with $n \ge 3$. Let $x \in X$ be a closed point. Consider the map given by projection from $x$: $$\phi: X \dashrightarrow \mathbb{P}^{n-1}$$ Suppose that ...
- 1,122
2 votes
1 answer
271 views
Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4. Near the end of section 3, ...
- 279
1 vote
0 answers
90 views
Discrepancy of general element of linear system
Let $X$ be a normal scheme and $|D|$ a linear system on $X$. In "Singularity of Minimal Model Program" by Janos kollar p249, it says, If $X$ is a variety over $\mathbb{C}$, and $E_j$ ...
- 368
0 votes
0 answers
55 views
Bôcher's theorem for singularities on the boundary
Let $\Omega\subset\mathbb{R}^2$ be connected, open, bounded, and smooth. Suppose that $u\in C^0(\bar \Omega\setminus \{0\})\cap C^2(\Omega\setminus\{0\})$ is harmonic and positive in $\Omega$. If $0\...
- 318
1 vote
0 answers
62 views
Does every holomorphic map admit a stratified submersion?
Given a map (germ) $g:(\mathbb{C}^{n+k},0)\rightarrow (\mathbb{C}^k,0)$, are there stratifications that make it a stratified submersion? By stratified submersion I mean a map that has stratifications ...
- 363
2 votes
0 answers
151 views
Punctured neighbourhood of quotient singularity is not simply connected?
Let $X$ be a variety (irreducible, normal) over a field $k$ which is algebraically closed with characteristic $0$. Suppose that $X$ has only one singular point $p\in X$, so $Y:=X\setminus p$ is smooth....
- 131
2 votes
1 answer
192 views
Blow up of terminal singularity and canonical singularity
A normal singularity $(X,x)$ over a field $k$ is terminal (resp. canonical) if $(i)$ it it is $\mathbb{Q}$-Gorenstein. and $(ii)$For any resolution of singularity $F:Y\rightarrow X$, $K_Y-f^*K_X>...
- 368
1 vote
0 answers
71 views
Making sense of constant scalar curvature metric horns
Suppose we have a compact oriented surface $S$ and we remove a point $p$ on it. We could consider a neighboorhood $U$ of the puncture $p$, so that the points in this neighboorhood are described by ...
1 vote
0 answers
77 views
Concerning the definition of a class of functions introduced by Nilsson
In the paper 'Some growth and ramification properties of certain integrals on algebraic manifolds' by Nilsson we find the following definitions: My question is how does one prove the remark "It ...
- 398
2 votes
0 answers
77 views
Smooth vs. topological: foliation into closures of orbits
Consider a (partial) map $f\colon X → X$ and the maximal closures of orbits of $f$ (i.e., closures of orbits which are not contained in larger closures of orbits). Assume that $X$ is foliated into ...
- 485
1 vote
0 answers
146 views
About the definition of cDV singularity
M. Reid defines cDV singularity as follow in his paper "CANONICAL 3-FOLDS" A point $p\in X$ of a 3-fold is called a compound Du Val point if for some section H throgh $P$, $P\in H$ is a Du ...
- 368
2 votes
0 answers
146 views
Finiteness of rational double point
Let $(R,\mathfrak{m })$ be a three dimensional complete local ring over a field $k$ of arbitrary characteristic and let $f\in R$ and $R/f$ is a rational double point. My question is as follows. Are ...
- 368
7 votes
0 answers
407 views
Is every normalization a blowup?
I asked this at math.stackexchange, but received no reply. Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example. ...
2 votes
0 answers
94 views
Estimating the growth rate around singular points of the analytic continuation of functions of Nilsson class defined by an integral
In Lemma 8 of the paper "Constant terms in powers of a Laurent polynomial" (by J.J. Duistermaat and Wilberd van der Kallen) the exponent $\alpha$ in the asymptotic expansion of a function of ...
- 398
1 vote
0 answers
67 views
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 ...
- 343
1 vote
1 answer
190 views
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 $4$ ...
- 2,467
8 votes
0 answers
667 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 ...
- 162k
5 votes
1 answer
586 views
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-...
- 26.3k
1 vote
0 answers
220 views
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 ...
- 73
3 votes
0 answers
118 views
Perturbation method for time-periodic singular system of ODEs
I am studying a problem arising in physics, and I managed to simplify it to a differential system (initial value problem) of the form: $$ \begin{cases} \dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
- 131
6 votes
0 answers
241 views
Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$
In my research I encountered automorphisms of the ring of convergent power series $$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$ which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
- 1,356
0 votes
0 answers
145 views
Is any singularity a subgerm of $(\mathbb{C}^n, 0)$?
I am studying singularity theory. I have often come across, in the literature, the sentence which says "let $(X,0) \subset (\mathbb{C}^n,0)$ be a singularity". Here a singularity is a ...
- 369
2 votes
0 answers
109 views
Continuous invariants of singularities in the Thom-Mather theory of deformations
I have been reading through Arnold et al.'s Singularities of differentiable maps to have an understanding on Arnold's theory of deformations of wave fronts. His theory is similar to the Thom-Mather ...
1 vote
0 answers
75 views
Characterization of the Picard's condition for integral equation
Picard's condition (Thm. 15.18, Kress et al. 1989) is essential to study the existence of a solution of a Fredholm integral equation of the first kind. Specifically, consider (the univariate case) the ...
- 153
6 votes
0 answers
244 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 ...
- 380
2 votes
0 answers
269 views
A question about the regularity of the Schrödinger equation
While reading the article [1], I noticed I don't understand part of the proof of regularity. For the Schrödinger eigenvalue problem, \begin{cases} -\Delta u+Vu=\lambda u, &\text{in } \Omega \\ \...
- 31
2 votes
1 answer
208 views
Why the following quasi isomorphism implies the morphism to be a resolution (a step in the paper A characterization of rational singularities)
I was reading the paper A Characterization of Rational Singularities by Professor Kovács. The main theorem is stated as follows: THEOREM 1. Let $\phi: Y \rightarrow X$ be a morphism of varieties over ...
- 225