Questions tagged [singularity-theory]
Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.
579 questions
2
votes
1
answer
157
views
Hyperplane sections of Gushel-Mukai threefolds
Let $X$ be a Gushel-Mukai threefold, and $S \in |\mathcal{O}_X(H)|$ a hyperplane section of $X$. How are possible hyperplane sections classified?
I know that if $S$ is a generic hyperplane section, ...
- 355
1
vote
1
answer
118
views
The normalization of a semi normal complex surface germ is a holomorphic immersion?
Given a reduced and irreducible complex surface germ $(X,0)$ which is seminormal, consider its normalization
$$
n : (\overline{X},0) \longrightarrow (X,0).
$$
Is the normalization map $n : \overline{X}...
- 135
2
votes
1
answer
215
views
Commutativity of nearby cycle functors and closed immersions under transversality condition
Let $X$ be a smooth variety over $\mathbb{C}$,
$Z$ be a closed subvariety of $X$.
Let $f:X \to \mathbb{A}_{\mathbb{C}}^{1}$ be a regular function, $X_0 := f^{-1}(0)$.
Denote $Z_0 = X_0 \cap Z$ and $i :...
- 121
4
votes
1
answer
282
views
Does the intersection of a variety with rational singularities with a smooth variety have rational singularities?
Let $X \subset \mathbb{P}^n$ be a complex projective variety with rational singularities, and $Y$ a smooth subvariety in $\mathbb{P}^n$ that meets $X$ transversally, that is to say their tangent ...
3
votes
1
answer
197
views
The Intersection number on a complex noncompact variety
Let $X$ be a variety over $\mathbb{C}$ and $C \subset X$ be a proper curve. Let $L$ be a line bundle on $X$. Then, the intersection number $L \cdot C$ is given by $\deg L|_{C^\nu}$, where $C^\nu$ is ...
- 573
1
vote
0
answers
63
views
Nodal varieties with simple resolutions
I am interested in classes of nodal varieties whose minimal resolutions (i.e. the blowups at all of the nodes) are of a well-understood form. For instance, the blowup of a nodal quartic surface is K3. ...
- 1,135
4
votes
0
answers
225
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 ...
- 573
0
votes
0
answers
116
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
192
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 ...
- 2,577
3
votes
1
answer
232
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 ...
- 2,577
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
263
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 ...
- 759
3
votes
0
answers
219
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
215
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
174
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
475
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}_{\...
- 313
3
votes
0
answers
86
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
333
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
259
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
106
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
241
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_*...
- 1,115
3
votes
1
answer
244
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, ...
- 1,691
1
vote
0
answers
121
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
267
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 ...
- 169
1
vote
0
answers
111
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. ...
- 1,135
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 ...
- 1,135
2
votes
0
answers
194
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
117
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
690
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
126
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
275
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
96
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 $E_j$ ...
- 378
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
152
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
201
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>...
- 378
1
vote
0
answers
75
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
84
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
81
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 ...
- 499
1
vote
0
answers
160
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 ...
- 378
2
votes
0
answers
151
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 ...
- 378
7
votes
0
answers
417
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
98
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
68
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 ...
- 363
1
vote
1
answer
191
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 $4$ ...
- 2,577
9
votes
0
answers
753
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 ...
- 164k
5
votes
1
answer
593
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-...
- 26.4k
1
vote
0
answers
230
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
120
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:
$$
\begin{cases}
\dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
- 131
6
votes
0
answers
245
views
Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$
In my research I encountered automorphisms of the ring of convergent power series
$$\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
146
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