The singularity-theory tag has no usage guidance.

**6**

votes

**1**answer

307 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 ...

**3**

votes

**2**answers

218 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 $A_2$ ($x^2+y^2+z^3=0$) ...

**3**

votes

**2**answers

250 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 ...

**10**

votes

**1**answer

264 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 ...

**2**

votes

**0**answers

85 views

### 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 ...

**6**

votes

**1**answer

150 views

### 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
$$ ...

**4**

votes

**1**answer

224 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. ...

**1**

vote

**1**answer

84 views

### 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 ...

**0**

votes

**1**answer

115 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 ...

**4**

votes

**2**answers

387 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 ...

**5**

votes

**1**answer

207 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 $\mathbb{R}^n$, ...

**4**

votes

**2**answers

225 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 ...

**1**

vote

**1**answer

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 ...

**1**

vote

**0**answers

90 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 ...

**8**

votes

**2**answers

895 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, ...

**10**

votes

**0**answers

244 views

### 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 ...

**2**

votes

**2**answers

309 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 ...

**1**

vote

**2**answers

274 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?

**8**

votes

**1**answer

573 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 ...

**1**

vote

**0**answers

84 views

### 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 ...

**3**

votes

**0**answers

316 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 ...

**9**

votes

**1**answer

336 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 ...

**8**

votes

**2**answers

1k 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
...

**5**

votes

**0**answers

470 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 ...

**6**

votes

**1**answer

530 views

### 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 ...

**3**

votes

**1**answer

165 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 ...

**0**

votes

**1**answer

457 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
...

**0**

votes

**1**answer

85 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, ...

**4**

votes

**0**answers

508 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 ...

**3**

votes

**1**answer

166 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 $X$ ...

**1**

vote

**1**answer

378 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 ...

**2**

votes

**1**answer

403 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 ...

**2**

votes

**2**answers

401 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?

**2**

votes

**1**answer

388 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 ...

**5**

votes

**1**answer

239 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 ...

**2**

votes

**1**answer

305 views

### 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 ...

**1**

vote

**1**answer

171 views

### 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 ...

**2**

votes

**1**answer

516 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 ...

**3**

votes

**0**answers

128 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 ...

**2**

votes

**2**answers

283 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 ...

**1**

vote

**0**answers

320 views

### Explicit Computations of Dynkin Diagrams of Isolated Singularities

Let $f$ be a complex polynomial with an isolated singularity at the origin. Take a Morse deformation $\tilde{f}$, and consider the braid group action on the set of distinguished bases of vanishing ...

**3**

votes

**0**answers

215 views

### Good covers on complex algebraic varieties with normal crossings singularities

Let $X$ be a topological space. A good cover on $X$ is an open cover such that all finite non-empty intersections are contractible. It is a theorem of Hironaka that (complex) algebraic sets admit ...

**1**

vote

**1**answer

433 views

### triple point singularity

Assume a complex surface $X$ admits a fibration structure over $\mathbb{CP}^1$ with some singular fiberes. Are there explicit examples of such surfaces with triple point singularity?

**3**

votes

**1**answer

311 views

**47**

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**2**answers

2k views

### When is a singular point of a variety smooth?

If $X$ is a nonsingular algebraic (or analytic) variety over $\mathbb C$ or $\mathbb R$ then it is certainly $C^\infty$ over the reals.
The converse is false for a silly reason : in the real or ...

**5**

votes

**0**answers

309 views

### What is known about “singularity types” in the Murphy's Law sense?

In his "Murphy's Law" paper, Vakil gives a definition equivalent to the following:
The singularity type of a pointed scheme $(X,p)$ its equivalence class, under the following equivalence relation: ...

**9**

votes

**3**answers

584 views

### Singular fibers of generic smooth maps of negative codimension

This is in some sense a follow-up to my question on submersions.
Let $f\colon\thinspace M\to N$ be a generic smooth map between closed manifolds of dimensions $m$ and $n$. Assume that the codimension ...

**1**

vote

**0**answers

210 views

### Do deformations of isolated hypersurface singularity naturally induce deformations of their divisors?

Let $0 \in V =(f=0) \subset \mathbb{C}^{n+1}$ be an affine variety with an isolated hypersurface singularity at the origin for $n \ge 3$.
Let $0 \in D=(x=f=0) \subset V$ be a divisor with only ...

**5**

votes

**0**answers

240 views

### Fixed point sets that carry topology

Let $M$ be a closed smooth manifold. A generic diffeomorphism $\phi: M\rightarrow M$ has non-degenerate fixed points, i.e. the intersections of its graph with the diagonal in $M\times M$ are all ...

**2**

votes

**1**answer

198 views

### Controlling singularities on log mmp

Suppose all my varieties are complex threefolds $X\rightarrow Y$ over some smooth base curve germ $Y$. We can assume the fibres are Del Pezzo surfaces with generic smooth fibre.
If I do (relative) ...