Questions tagged [singularity-theory]
Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.
554 questions
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resolution for the du Val's $(A_3)$-singularity
For the $A_m$-singularity, it can be viewed as the singular part of $\mathbb{C}^2/\mathbb{Z}_m$. The action of $\mathbb{Z}_m$ on $\mathbb{C}^2$ is defined as following
$$
\bar{1} \cdot (z,w) = (z e^{\...
- 167
1
vote
0
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328
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Cohomology of a structure sheaf of a normal affine variety
I can't find the reference for the following fact:
Let $X$ be an affine variety and let $Y$ be its smooth resolution. $H^0(X,\mathcal{O}_x)=H^0(Y,\mathcal{O}_Y)$ if and only if $X$ is normal.
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7
votes
1
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317
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Holomorphic vector fields tangent to a hypersuface singularity
Let $(V,0)\subset (\mathbb{C}^n,0)$, $n\geq 3$ a (germ of) hypersurface given by $V =\{f=0\}$, $f$ a germ of holomorphic function. A (germ of) holomorphic vector field $X$ on $(\mathbb{C}^n,0)$ is "...
- 502
1
vote
1
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408
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Jacobian ideal regular sequence
Let $f(x,y) \in \mathbb C\{x,y\}$ be a holomorphic function-germ at zero. Let $f_x, f_y$ denote its partial derivatives. What is the proof of the following statement?
If the $\mathbb C$-algebra $\...
- 81
4
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1
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330
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example of quintics with 5 ordinary triple point
I know we can bound the triple point on quintics in cp^3 by 5. But how to write down quintics with 5 ordinary triple point (here are simple elliptic singularity)explicitly?
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9
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1
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Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?
Let $X$ be an affine variety. Let $Y$ be smooth and let the map $f\colon Y\rightarrow X$ be proper birational. We will call $Y$ a smooth resolution of $X$.
Do the cohomology groups $H^i(Y,\mathcal{O}...
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2
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1
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How can I compute the mixed hodge structure for three copies of $\mathbb{P}^1$ intersecting at one point?
I know there is a spectral sequence for a variety with normal crossing singularities $X$ which gives a tool for making the computation of the mixed hodge structure computable. How can I compute the ...
- 1,716
4
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0
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182
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Kuranishi family and smoothing of Calabi-Yau n-fold
Consider $X$ be a Calabi-Yau n-fold with at
most one ordinary double point singularity. Suppose $X$ is smoothable. Then it is known that the Kuranishi family of $X$ is a smoothing of $X$.
Now, ...
- 81
4
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0
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218
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Example of a non-algebraic singularity II
In an answer of this MO question, Frank Loray constructed an example of analytic singularity which is not algebraic. On the other hand, as I learned from one of Joël's comments in that question, ...
- 1,829
2
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0
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classification of simple singularities for the henselian of the plane Zariski local ring at the origin
Let $A=\mathbb{C}[x, y]_{(0, 0)}=\{\frac{f(x, y)}{g(x, y})\in \mathbb{C}(x, y)\ |\ g(0, 0)\neq 0\}$ be the Zariski local ring at the origin. Then the henselisation $A^h$ of $A$ is the ring of formal ...
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0
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68
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Poisson cohomology of germfied Poisson structures in dimension two
Let $f(x, y)$ be a smooth function in the real case or a holomorphic function in the complex case. Denote $\pi=f（x, y)\frac{\partial}{\partial x}\wedge \frac{\partial}{\partial y}$ be the ...
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3
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0
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99
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Normal form of volume functional about a minimal surface
Let $S$ be a closed manifold and $(M,g)$ be a Riemannian manifold. Minimal submanifolds are by definition the critical points of the volume functional
$$F: \mathcal{Imm}(S,M) \to \mathbb{R} \qquad \...
- 5,824
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352
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smoothing of isolated surface singularity
I want to know when an isolated surface singularity can be smoothed, especially for log canonical isolated surface singularity. Is there any good reference. Thanks in advance.
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0
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300
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How to make a product of polynomials irreducible?
Let $p(x,y),q(x,y)\in \mathbb{Q}[x,y]$. Assume that $p(0,0)=q(0,0)$. Is it generally possible to find a polynomial $r(x,y)\in\mathbb{Q}[x,y]$ irreducible in $\mathbb{Q}[x,y]$ such that $\mathbb{R}[x,y]...
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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 ...
- 8,071
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0
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86
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Action of the monodromy on the cycle made of the real points
Let $f : \Bbb C^n \to \Bbb C$ be a polynomial function with real coefficients.
Let $X_t = f^{-1}(t)$ denote the fiber above some $t \in \Bbb C$. Let assume that the set of real points of $X_t$, for $t ...
- 1,044
3
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0
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447
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simple elliptic surface singularity
Suppose X is a one dimension torus and L is a line bundle over X, I think one class of log canonical surface singularity comes from the contraction of one elliptic curve from the total space L. My ...
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2
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0
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296
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An approach for Singular Hermitian-Einstein metric
Motivation: If we extend Hitchin-Kobayashi correspondence along holomorphic fibre space such that each vector bundle $E_s$ of fibers $X_s$ be stable then for finding canonical twisted Hermitian-...
user21574
2
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0
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266
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Blow-up along singularity of the degeneracy locus
Let $X$ be a smooth, projective variety, $E$ a rank $r$ locally free sheaf on $X$. Fix a closed embedding $i:X \hookrightarrow \mathbb{P}^N$ and denote by $\mathcal{O}_X(m)=i^*\mathcal{O}_{\mathbb{P}^...
- 1,981
2
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0
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214
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log canonical surface singularity
For a log cannonical surface singularity, I guess there is classification about the configuration of the exceptional divisor of its minimal resolution. I wonder if there is some specific example or ...
- 623
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0
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82
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Singularities of fibrations 2
This question is related to my previous question:
Singularities of fibrations
Assume that $X$ is a complete intersection irreducible $3$-fold in a product of projective spaces. So that $X$ is ...
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4
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1
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327
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Singularities of fibrations
Let $f:X\rightarrow \mathbb{P}^2$ be a fibration, here $X$ is a projective variety of dimension three.
Assume that there exixts a smooth curve $C\subset\mathbb{P}^2$ such that for any $p\in\mathbb{P}...
- 8,998
2
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1
answer
167
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Singularities of $3$-folds
Let $X,Y,Z$ be projective $3$-folds. Assume that $Y$ is smooth and $Z$ is smooth and Fano. Moreover, assume that there is a generically finite morphism $f:Y\rightarrow Z$ admitting a factorization $f=...
user97096
5
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0
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319
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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 ...
- 1,981
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2
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203
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Asymptotic behavior of the ratio between the largest two singular values of product of i.i.d. random complex matrices
Let $A_n$ be the matrix product of $n$ i.i.d. N-by-N random complex matrices. The matrix distribution is not fixed and can be tuned to suit specific solution if needed, as long as it's not too "...
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3
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0
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Resolving complete intersections of quadrics with singularities
Suppose that $X$ is a complete intersection of quadrics in $\mathbb P^n_{\mathbb C}$. Is there some straightforward procedure to resolve the singularities of $X$?
For example, can one stratify ...
- 14.3k
4
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1
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505
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Lefschetz hyperplane section theorem for intersection homology
Let $X$ be a smooth, projective variety and $Y \subset X$ be a hyperplane section (possibly singular) of $X$. Suppose that the dimension of $X$ is $n$. Is it true that for any $k<n-1$, the induced ...
- 2,323
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0
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203
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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 $...
- 21.8k
3
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0
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131
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Classification of faithfully flat morphisms between formal power series
Let $\mathbb{C}[[z_1,\dots,z_n]]$ denote the algebra of formal power series.
I am interested in faithfully flat morphisms
$$Spec(\mathbb{C}[[z_1,\dots,z_m]])\to Spec(\mathbb{C}[[z_1,\dots,z_n]]),\, m\...
- 21.8k
25
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1
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4k
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Can the constant rank theorem for smooth manifolds be generalized to nonconstant rank?
The constant rank theorem says that
if $f\colon M→N$ is a smooth map whose rank equals some fixed $k≥0$ at any point of $M$, then, locally with respect to $M$ and $N$, the map $f$ assumes the easiest ...
- 37.8k
4
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2
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737
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Whitney Conditions vs Equisingularity
In studying singular spaces, it is often important to pick an appropriate stratification which encodes the singularity structure. One class of such stratifications are called "Whitney stratifications" ...
- 1,073
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0
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229
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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 ...
- 1,981
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0
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247
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H-principle for smoothing
I'm trying to find algebraic embedding of singular curves into projective varieties that can be smoothed symplectically but not algebraically.
It's not hard (e.g. using the methods in Hartshorne-...
- 1,981
2
votes
1
answer
512
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Grothendieck duality for resolution of singularities
I would like to know a reference for Grothendieck duality in a resolution of singularities. More precisely, let $Y$ be a normal, Gorenstein variety with finite quotient singularities, and suppose that ...
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3
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1
answer
736
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semi-log canonial singularities is an open condition?
Let $f:X\to \mathbb \Delta$ be family of projective varities which fibers are smooth, we know central fiber can be singular and may not be mild. Kollar introduced semi-log-canonical singularities to ...
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3
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1
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snc singularity is an open condition?
If $π : X → S$ is a family of complex projective
varieties, such that $X_0 := π^{−1}(0)$ has simple normal singularities in $X$, then all the general fibers $X_t$ have snc singularities at worst?
- 135
2
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0
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98
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Singularities of $Spec(Sym^* E^{\vee})$ for $E$ a coherent sheaf
Let $X$ be a smooth complex algebraic variety, and $\mathscr{E}$ a torsion-free coherent sheaf on $X$.
Which type of singularities can the total space $\mathrm{Tot}(\mathscr{E}):=\underline{\mathrm{...
- 23.4k
1
vote
0
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74
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Simple question about surface singularities
Given $\epsilon \in (0,1)$, is it possible to find two finite familes $\mathcal{F}$ and $\mathcal{P}$ of weighted graphs, such that the weighted graph of the minimum resolution of any $\epsilon$-klt ...
- 1,595
4
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1
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401
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Bernstein's theorem
In the book "An introduction to the theory of local zeta functions" prof. Igusa presents Bernstein's theorem as follows:
Let $K_0$ be a field, and write $K=K_0(s)$. Let $f\in K_0[x_1,\dots,x_n]\...
- 2,788
6
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1
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617
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Advantage of discrepancy
In the definition of Minimal model of projective variety, some authors use of discrepancy, and some others omit this condition. I am wondering to know the advantage of discrepancy In the definition of ...
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2
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1
answer
559
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quotient singularities
Let $X$ be a relaively compact projective variety and has only quotient singularities then for any n-form $\Omega$ , $$\int_{X_{reg}}\Omega\wedge \bar \Omega$$ is bounded? what about the converse
- 99
5
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0
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727
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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 ...
- 1,462
5
votes
2
answers
676
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Log canonical counterexample to Kawamata-Viehweg vanishing
I found in the literature that, in characteristic 0, Kodaira vanishing holds for log-canonical pairs. On the other hand, the usual statement for Kawamata-Viehweg vanishing talks about a klt pair $(X,\...
- 625
6
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1
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756
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What tools can I use to compute the cohomology of the fibers of a Lefschetz Pencil?
I'm learning about Lefschetz pencils and vanishing cycles and have looked at a few sources:
http://www.math.purdue.edu/~dvb/preprints/sheaves.pdf
http://www3.nd.edu/~lnicolae/Morse2nd.pdf
Voisin's ...
- 1,716
4
votes
1
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226
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Is the factorial cDV-singularity $T_1^2 + T_2^3 + T_3^4T_4$ any quotient of any affine space by any group?
the reason for my question is the following:
the two-dimensional canonical singularities are the ADE-singularities, which all are quotients of either affine space or another ADE-singularity by finite ...
- 161
1
vote
1
answer
417
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Central fibre singularities
Let $f:X\to Y $ be a proper surjective holomorphic fibre space where $X,Y $ are projective varieties.
If the central fibre $X_0$ has at worst log terminal singularities,
then can we say that all ...
user21574
1
vote
1
answer
203
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Does a moving family of lines through a fixed point produce a singularity?
This is just a feeling that I had and I am curious if it is totally wrong or true to some extent.
Let $X\subseteq \mathbb{P}^r$ be an integral hypersurface of degree $r-1$, which is not a cone. In ...
- 325
2
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0
answers
114
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Orbit spaces of Coxeter groups and singularities
I have often seen in the literature the statement that the orbit spaces of irreducible finite Coxeter groups are equivalent to unfoldings of singularities.
For instance, taken from Dubrovin, ...
- 1,821
4
votes
2
answers
1k
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Kähler forms arising as the curvature form of a singular metric on a line bundle
The Fubini-Study metric on complex projective space $\mathbb{P}^n$ is a smooth metric $h = e^{-\phi}$ on the line bundle $\mathcal{O}(1)$ and it is a standard calculation to check that its curvature ...
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1
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1k
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Are rational surface singularities $\mathbb{Q}$-Gorenstein?
I know that, in general, rational singularities are not necessarily $\mathbb{Q}$-Gorenstein. So I ask:
is there any positive result in this direction known for surfaces?
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