# Questions tagged [singularity-theory]

The singularity-theory tag has no usage guidance.

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### Existence of meromorphic 2-forms over normal surface singularities

Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^...

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### Cohomology of real analytic coherent sheaves

Let $M$ be a real analytic variety
(if someone is concerned about distinction between
"real analytic spaces" and "real analytic varieties"
in real analytic geometry, let's assume that $M$
is both "...

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### An upper bound for the number of singularities of a transversal vector field isometric to the zero field

Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$.
A smooth vector field $X:M \to TM$ is called a transversal vector field if $X(M)$ is transverse ...

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### Equations needed to define a normal complex surface singularity

This questions is highly related with this other question of mine: Irreducible surface singularity that is not a local set-theoretical complete intersection I just thought that a different look at the ...

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### Relationship between Hilbert-Samuel multiplicity and polar multiplicity

Let $f \in \mathbb{C}[[x,y]]$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $e_f$ of $f$ is given as follows:
$$e_f = \lim_{s \to \infty}\frac{1}{s} \cdot \...

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### Irreducible surface singularity that is not a local set-theoretical complete intersection

I have been looking for a criterion for the germ of an irreducible complex surface singularity $(X,x)$ to be a set-theoretical complete intersection.
A germ $(X,x)$ of an isolated complex singularity ...

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### Smoothings over a real interval

I am asking if somebody knows if the following kind of objects are studied somewhere or if there is some kind of obvious obstruction for them to appear.
Let $(X,0) \subset \mathbb{C}^n$ be a germ of ...

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### Smoothings of isolated non-irreducible surface singularities

Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing.
Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not ...

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### On Remmerts reduction

Let $(X,0)$ be a normal surface singularity. An let $\pi: \tilde{X} \to X$ be the minimal resolution. Now, we can apply a result of Oliveira (exploiting previous work by Laufer) and obtain a 1-...

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### Slow and fast forming singularities of the mean curvature flow

Let $M \times [0, T) \to \mathbb{R}^{n+1}$ be a mean curvature flow and let $T$ be a singular time. Let $A$ denote the second fundamental form.
We have a type I singularity if
$$
\max_{p \in M} |A(p,...

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### The homology groups of the smooth locus of a singular variety

Let $X$ be a complex irreducible variety and denote its smooth locus by $X^{smooth}$. I would like to know what can be said about the induced maps $H_i(X^{smooth};\mathbb{Q})\rightarrow H_i(X;\mathbb{...

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### The surface singularity $x^4=yz$

Sorry if this question is a bit broad. I would like to have examples of papers which have studied the surface singularity
$$x^4=yz,\quad(x,y,z\in\mathbb{C}).$$
I am trying to get a feel about what is ...

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### Can we extend a logarithmic form to some appropriate compactification?

Given some simple normal crossings divisor $D$ on a complex manifold $X$, which is not assumed to be compact. Given a form $\omega\in H^0(X,\Omega_X^1(\log D))$, when is it possible to find a ...

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### Is canonical model always with canonical singularity

Let X be a smooth variety, take the proj of canonical ring of X and denote it by Y. Is Y always a canonical variety? I know it's true for general type variety. Thank in advance.

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### Singularities of a central fibre of a flat family of smooth surfaces

Suppose I have a one parameter flat family of complex surfaces (regular, of general type) whose general fibre is smooth. Is it possible for the central fibre to have singularities which are not ...

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### Deformation sublevel sets of functions which preserve boundary

I'm interested in proving the following fact, which seems to naturally arise from gradient flow deformations, but appears to be a bit tricky.
Consider a smooth family
$$f_s : M \to \mathbb{R}, \quad ...

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### Generalized Sard's lemma

Let $f: X \to \mathbb{R}$ be a $C^{1,1}$ (that is $C^1$ with Lipschitz differential) function on a manifold $X$. Suppose that $f$ is smooth at all points of a subset $C \subset \text{Crit}f$ of ...

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### A paradox on the deformation of singularities

Setup: $\pi: \mathcal X \to C$ is a flat morphism from a germ of a smooth curve $(C, o)$. Suppose the special fiber $\pi^{-1}(o) := \mathcal X_o$ has a certain class of singularities, one wants to ...

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### Pushforward of structure sheaf on quotient surface singularity

Let $f:\mathbb{C}^2 \to \mathbb{C}^2/G$ be a quotient surface singularity. What properties should $G$ have such that the pushforward $f_*\mathcal{O}_{\mathbb{C}^2}$, of the structure sheaf $\mathcal{O}...

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### Desingularization of the zero section of $TM$ as the manifold of singularities of the geodesic flow

However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities"...

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### Can nonflat deformations of singularities always produce Cohen-Macaulay rings?

To make the question in the title precise, let me phrase it like this. Consider a complete local ring
$$ A := \mathbb{C}[[x_1, \dotsc, x_n]]/(f_1, \dotsc, f_m) $$
and, for definiteness, assume that $...

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### Singularities of quotient of a vector bundle by a lattice

Let $V$ be a complex vector bundle of rank $g$ over an open unit disc $\Delta$ and $H$ be a integral local system of rank $2g$ over $\Delta$ i.e., for every $t \in \Delta$, $H_t \cong \mathbb{Z}^{2g}$....

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### Enumerating the unfoldings of real singularities

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a germ of an isolated, real analytic singularity. Let $I$ be the ideal in $R = \mathbb{R}[x_1, \dots, x_n]$ generated by the components of $\nabla f$, and $Q = ...

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### Connections between eigenvectors after matrix multiplication

Suppose we have an M$\times$N complex matrix $H$ and its singular value decomposition $H=U\Lambda V^*$ and an N$\times$N covariance matrix $R_s$ with its eigendecomposition $R_s = U_s\Lambda_sU_s^*$. ...

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### Small contractions as blow ups

To improve my chances of getting answers/ comments I post my mathstackexchange question https://math.stackexchange.com/q/2808852/42548 also here.
I am trying to learn a bit about birational morphisms:...

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### Bertini-type theorem for reducible schemes

Let $X \subset \mathbb{P}^n$ be a reducible, projective subscheme. Assume that $X$ is reduced (meaning that every local ring is reduced i.e., does not contain nilpotent element). Denote by $S_d$ the ...

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### Example of maps between a smooth curve and a singular curve

I would like an example of maps between a smooth curve $C$ and a singular curve $B $, $f:C \rightarrow B$, where the genus $p_a(C)=p_a(B)$ and greater than or equal to 2.

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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 ...

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### On regularity of flat families over a DVR

Let $k$ be an algebraically closed field of characteristic zero and $R$ a discrete valuation ring over $k$. Let $\pi:X \to \mathrm{Spec}(R)$ be a flat, projective morphism such that the generic fiber ...

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### Normal form of functions $(x^2+y^2)^n+$ higher terms

By Morse lemma for any $C^{\infty}$ function $f$ on $\mathbb R^2$ with Taylor series $(0,0)$ starting with $x^2+y^2$ one can find local $C^{\infty}$ coordinates $(x',y')$ such that locally $f(x',y')=...

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### Singularities of the union of two smooth curves

I am looking for a reference that describes what are the possible singularities of a curve $C=C_1 \cup C_2$ which is the union of two smooth irreducible curves $C_1,\,C_2$ on a smooth surface.
Doing ...

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### Typical singularities of a map from $\mathbb{T}^{m}$ to $\mathbb{R}^{n}$

My question is as follows. Is there known general results on typical singularities (critical points) for smooth maps from $m$-dimensional torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\mathbb{Z}^{m}$ to $\...

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### Classification of singularities of plane curves of fixed degree (reference request)

We know the answers to some questions like What is the maximal number of singularities of (reduced) plane curves of degree $d$? for general $d$ (in this case $\tfrac{1}{2}d(d-1)$, obtained by $d$ ...

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### Frobenius structure for A_n singularities

I need to compute monodromy matrices $M(v)$, associated to a Frobenius structure for $A_n$ singularity with flat coordinates $v_1,\dots,v_n$, that is, $f(x)=x^{n+1}$. (due to Saito, Dubrovin etc.) ...

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### Reference for certain resolution of singularities formulation

I want to use the following resolution of singularity statment as found in Soule et al, Lectures on Arakelov Geometry, p. 40:
$Y$ is a separated algebraic variety of finite type over $\mathbb{C}$, $Z$...

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### How singular is the metric on an orbifold

I am reading some stuff on orbifolds. I am particularly interested in the metrics on orbifolds. The famous example of one orbifold is the "American football", which is $\mathbb{S}^2$ quotient by the ...

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### A question about explicit computations of discrepancies

The following is an explicit computation of discrepancies appeared in the book "Birational Geometry of Algebraic Varieties" (Page 126-127) in order to show certain type singularities are not Du Val. ...

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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 ...

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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^{\...

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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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### 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 "...

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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 $\...

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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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### 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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### 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 ...

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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, ...

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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, ...

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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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### 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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### 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 \...