The singularity-theory tag has no wiki summary.

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### Which isolated surface singularity comes from a -5 curve?

Define the surface $X$ to be the total space of $\mathcal{O}_{\mathbb{P}^1}(-5)$.
By contracting the exceptional curve in $X$, we get a surface with an isolated singularity. I am looking for the ...

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146 views

### Obstruction map for local singularities via tangent (Andre-Quillen) cohomology

Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent ...

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258 views

### Normal form for a holomorphic Morse function

Similarly to Morse lemma, a holomorphic Morse function can be written, near a critical point, as $W_1^2+W_2^2+...+W_n^2+C$ , for well chosen coordinates $W_1,W_2,...,W_n$. I want to cite this result ...

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227 views

### on a characterisation of regular D-modules

Let $X$ be a smooth variety over a field of characteristic zero. Let $M$ be in the derived category of holonomic $\mathcal{D}_{X}$-modules, $D^{b}_{h}(\mathcal{D}_{X})$.
We know that if we assume ...

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130 views

### Hypersurfaces with Gorenstein singular loci

Recall that a hypersurface $D$ in a complex manifold $X$ is called a free divisor if the Lie algebroid $\mathcal{T}_X(-\log D)$ of vector fields tangent to $D$ is locally free. This condition is ...

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56 views

### on Neron defect of smoothness for groups schemes

Let $G$ a semisimple simply connected group over $\mathbb{C}$.
Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$.
We consider $I_{\gamma}$ the group ...

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166 views

### Understanding maps from M to R^n, for n>dim M

I am interested in "approximating" smooth maps from a compact smooth manifold $M$ of dim $m$ into $\mathbb R^n,$ for $n>m,$ by "nice" maps, with properties similar to those of Morse functions. Of ...

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80 views

### cotangent complex for finite flat morphism and different ideal

Let a ring $A$, $F=A[X_{1},\dots X_{n}]$ and $B:= F/J$.
We suppose that we have a finite flat lci morphism $f:Spec(B)\rightarrow Spec(A)$.
To mesure the singularities of this map, Gabber-Ramero ...

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66 views

### semicontinuity of the conductor defined by Temkin

We say a principal pair $(X,\mathcal{I})$ where $X=Spec(A)$ is affine scheme and $\mathcal{I}=\tilde{I}$ where $I\subset A$ is a principal ideal generated by $\pi$ wich is a non zero divisor.
For a ...

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165 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
...

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257 views

### Iterated Milnor fibrations and Thom's a_f condition

Ok so there's a lot of litterature about nearby cycles functor since it was introduced by Grothendieck and Deligne but I couldn't find any clear answer to the following natural question:
Problem: Let ...

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205 views

### Surfaces in $\mathbb P^3$ with many simple isolated singularities

Could anybody help me with examples of surfaces $X\subset\mathbb P^3$ (projective, over $\mathbb C$) having many isolated singularities of the type $A_1$ ($x^2+y^2+z^2=0$) or $A_2$ ($x^2+y^2+z^3=0$) ...

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199 views

### Factoriality of one-nodal Calabi-Yau threefolds

Let $X$ be a projective Calabi-Yau threefold with a single ordinary double point at $x \in X$, and smooth elsewhere. Is $X$ necessarily factorial?
I suspect that the answer is "yes", for the ...

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241 views

### Analogue of singularity theory in other categories

Whitney, Thom, Mather, Arnold and others develoved the singularity theory of smooth maps.
Does there exist any analogue of this theory in the category of TOP or PL (or Lipschitz) maps?
I mean notions ...

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### Is it obvious that the defining conditions to obtain a particular singularity are well defined on the quotient space?

Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function
vanishing at the origin, with
the following properties:
$$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...

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### Is there an algorithm to find out the number of small solutions to a polynomial equation, when we vary all the coefficients?

Let $\Phi (z,t)$ be a polynomial given by
$$ \Phi(z,t) := z^n + A_{n-1}(t) z^{n-1} + \ldots + A_1(t) z + A_0(t).$$
Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation
$$ ...

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200 views

### Is Gorenstein singularity a locally analytic property

Let $X$ be a variety over $\mathbb{C}$, and $X^{an}$ be the analytic space associated to $X$. $X$ has Gorenstein singularity at $x \in X$ iff the local ring $\mathcal{O}_{X,x}$ is a Gorenstein ring. ...

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### Finite construction of lacunary functions using algebraic and certain analytic operations

Algebraic functions have a discrete set of singularities. Lacunary functions, e.g. $f(z)=\sum_{n=0}^\infty z^{2^n}$, have a continuum of singularities at every point of the boundary of their disk of ...

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106 views

### Is the space of degree $d$ curves with marked smooth points dense inside the space of curves with marked points?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d} $ be the space of nonzero
homogeneous degree $d$ polynomials in three variables upto scaling, where
$\delta_d = \frac{d(d+3)}{2} $
(basically degree ...

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360 views

### Is there an analogous concept for the degree of a map, when the spaces are singular?

Let $M$ and $N$ be two smooth compact, oriented manifolds and
$X\subset M$ an oriented submanifold of $M$ of dimension $k$
(not necessarily closed). Suppose in addition that $\bar{X}-X$ is contained ...

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192 views

### Are there general position results in singular algebraic sets?

Let $X$ be a real algebraic set, and let $Y \subset X$ be its singular set. In this question I'll focus on the analytic topology, so we can just imagine that $X$ is the zero set, in $\mathbb{R}^n$, ...

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181 views

### When is the intersection of an isolated normal singularity with a generic linear subspace through that singularity normal?

Suppose I have an affine subvariety $A \subset {\mathbb C}^N$ of dimension $n \geq 3$ which has an isolated singularity at $0$ (lets say for the sake of simplicity that it is non-singular everywhere ...

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92 views

### On solution of a recursion with rectangular matrices

Greetings to members here.
The question is how to calculate the solution $S(k)$ of the following recursive equation
$$J(k)S(k+1)J^{T}(k)=A(k)S(k)A^{T}(k)+R(k)$$
where $J$ and $A$ are rectangular not ...

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83 views

### How do I check whether an orbifold admits deformations?

(Cross-post from math.stackexchange, where it has received no attention.)
Orbifolds $\mathbb{C}^2/\mathbb{Z}_n$, given by the action $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$ with $\zeta$ a primitive ...

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579 views

### Whitney stratifications

Many results on characteristic classes of singular varieties (as well as other singularity-theoretic constructions) make use of a so-called "Whitney stratification" of the variety under consideration, ...

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### 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 ...

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259 views

### Bounds for the milnor number of a hypersurface singularity

I am having a hard time in finding an upper bound in terms of the degree and the dimension for the Milnor number of an isolated hypersurface singularity. I am mostly interested in surfaces on the ...

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259 views

### Producing $(-2)$ curves on a smooth surface

We know that blowing up a point on a surface produces a $(-1)$ curve. Is there any such standard techniques to produce $(-2)$ curves in a smooth surface?

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390 views

### Factoriality: local or global?

Let $X$ be an algebraic variety. I have read the following definitions:
$X$ is factorial if every Weil divisor on $X$ is Cartier.
$X$ is locally factorial if all its local rings are unique ...

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### Smoothing of a hyperquotient singularity

Let $f$ be a polynomial in $k$ complex variables, and suppose the affine variety $V$ given by $f = 0$ has an isolated singularity at the origin, but is otherwise smooth. Now assume that some cyclic ...

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289 views

### Blowing up a projective surface

Let $X$ be a smooth degree $d$ ($d>5$) surface in $\mathbb{P}^3$. We now blow up $X$ at a point, embed it in some projective space, and and consider a projection of it into $\mathbb{P}^3$. The ...

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283 views

### Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...

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683 views

### “Arithmetic genus” of a plane curve singularity.

I believe that the following questions are very basic, but I don't know how to get a reference.
Consider a curve in the plane $C\in \mathbb C^2$ with a singularity at $0$ and suppose it is
...

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347 views

### 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 ...

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### Factoriality vs $\mathbf{Q}$-factoriality for threefolds hypersurfaces with isolated singularities

Let $X \subset \mathbf{P}^4$ be a complex threefold hypersurface with isolated singularities.
We denote as usual by $\textrm{Cl}(X)$ the group of Weil divisors modulo linear equivalence and by ...

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147 views

### Counting nodal singularities on a surface

How many lines in $\mathbf{P}^5$ passing through a fixed point $p$ meet in at least two points a fixed smooth surface $S$ given by the intersection of three quadrics?
Or equivalently, calling $T$ the ...

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444 views

### Does the closure of a smooth algebraic always define a homology class?

Let $X\subset \mathbb{C} \mathbb{P}^{N}$ be a smooth,
algebraic (locally closed) complex
submanifold of $\mathbb{C} \mathbb{P}^N$
of complex dimension $k$. More concretely, $X$ is of the
...

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84 views

### Question regarding closure of sets defined by the vanishing of holomorphic functions

Consider the following subsets of $\mathbb{C}^n$ given by
$$ X := \{x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 \} $$
$$ Y := \{ x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 \} $$
where $f, ...

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370 views

### 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 ...

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144 views

### local fundamental group of elliptic singularities

Is the local fundamental group of an elliptic singularity virtually solvable ? Here (the terminology is sometimes divergent) an elliptic singularity is a (germ of) normal surface $(X,x)$ such that $X$ ...

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323 views

### Doubt about normality and rational singularities

In M. Reid Canonical 3-folds I found this proposition:
If $\phi:Y\rightarrow X$ is a proper morphism with both $X$ an $Y$ normal and such that $f$ is étale in codimension 1 then
1) if $X$ has ...

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344 views

### Applications of Slope Stability

Ross and Thomas developed slope-stability of $(X,L)$ where $X$ is an $L$-polarised variety and $L$ is an ample line bundle, as an obstruction to K-stability of $(X,L)$.
DISCLAIMER: (Forgive me if I ...

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292 views

### KLT singularities are quotient in codimension 2

I have read that if a variety $X$ has KLT singularities, then it has quotient singularities in codimension 2.
Do you know a proof (or where can I find a proof) of this?

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363 views

### Asymptotics on implicit function

We consider the asymptotics of the coefficients of generating function $y(x)$,
which is defined by the implicit function $y= F(x,y)$.
Let $F(x,y)$ be a rational function in $x$ and $y$, such that ...

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238 views

### Measuring contact between algebraic varieties

I have two regular surfaces in three space, both of which are given by an equation. I would like to measure the contact between the two surfaces using only their equations. Usually, one would find a ...

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### T^i functors are isomorphic for analytically isomorphic isolated singular points

I've been having trouble proving the following:
Let $B$ and $B'$ be local rings, essentially of finite type over $k$, having isolated singularities at the closed points. Suppose that they are ...

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### Adding singular equations to a smoothing of a hypersurface singularity

Let $f \in \mathbb{C}[x,y,z]$ be a polynomial which defines an isolated singularity $0 \in D:= (f=0) \subset \mathbb{C}^3$.
Assume that $\mathcal{D}:= (f+tx =0) \subset \mathbb{C}^3 \times ...

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480 views

### General position argument

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be ...

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118 views

### Equivalence of Level Sets

Consider the zero level set of $f : \mathbb{R}^3 \to \mathbb{R}$, where $0$ is a regular value. Consider also the space of planes passing through the origin, i.e. $\mathbb{RP}^2$. For a fixed plane $P ...

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269 views

### Does a generic curve inside the space of curves with a node at a specific point have only finitely many nodes?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be ...