The singularity-theory tag has no usage guidance.

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votes

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710 views

### Can one prove vanishing of higher direct images fiber-wise?

Let $\pi:X\to Y$ be a proper map of algebraic varieties (over $\mathbb C$) which is a bi-rational equivalence.
are the following statements equivalent?
The derived direct image of $O_X$ is $O_Y$.
...

**1**

vote

**1**answer

181 views

### How can one bound 'the lower perverse degree' for a constant sheaf on a variety that is smooth in high codimension?

Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of ...

**7**

votes

**0**answers

451 views

### Physicists Euler number conjecture

Physicist's Euler number conjecture says:
If $G \subset SL(n,\mathbb{C})$ is a finite group, $X=\mathbb{C}^n/G$ is the quotient space and $f:Y \rightarrow X$ a crepant resolution (always exists for ...

**10**

votes

**2**answers

1k views

### Giant Rat of Sumatra singularity

I would be grateful for explanations of the issues raised in any
of these three questions, or pointers to the relevant literature
(now updated with answers):
How did a particular singularity come ...

**6**

votes

**1**answer

564 views

### Blowing-up an ordinary double point, then contracting the exceptional locus to a curve

Let $X\subset\mathbb P^4$ a projective hypersurface with an ordinary double point at $o\in X$.
Blow-up $\mathbb P^4$ at $o$ and let $E\simeq\mathbb P^3$ the exceptional divisor of this blow-up. ...

**7**

votes

**2**answers

452 views

### Resolution of singularities for flat families.

Is there a resolution of singularities for flat families?
More precisely, if $X \rightarrow \mathbb{A} ^n$ is a flat map, does there exist a map $Y \rightarrow X$ such that, for every $p \in ...

**9**

votes

**2**answers

2k views

### construct the elliptic fibration of elliptic k3 surface

Hi all,
As we know, every elliptic k3 surface admits an elliptic fibration over $P^1$, but generally how do we construct this fibration? For example, how to get such a fibration for Fermat quartic?
...

**4**

votes

**1**answer

295 views

### Property of singularity

Let $X$ be an algebraic variety, $S \subset X$ its singular locus, and $x \in S$. Say that $x$ is good if for any neighborhood $U$ of $x$, any top differential form $\omega$ on $U \setminus S$ and ...

**4**

votes

**2**answers

555 views

### Vanishing associated to a resolution of singularities

Let $\pi: V\to W$ be a resolution of singularities, let $E \subset V$ be the exceptional divisor, and let $F$ be a coherent sheaf such that $R^i\pi_*F=0$ for $i>0$.
Can we conclude that ...

**6**

votes

**2**answers

1k views

### Places to learn about Landau-Ginzburg models

Here is what I know about Landau-Ginzburg models:
It is an important player in the story of mirror symmetry.
It involves "potentials" which are functions of complex varibles, which have isolated ...

**6**

votes

**2**answers

408 views

### How can we find a surface with a given singularity?

I was surprised the first time I learned that a quintic plane curve can have an $A_{10}$ singularity i.e $x^2+y^{10}$. I am wondering if there is something about that phenomenon: Given a singularity ...

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votes

**2**answers

718 views

### Implicit function theorem at a singular point?

Let $F:\mathbb{R}^2 \rightarrow \mathbb{R}$ be three times continuously differentiable in some open neighborhood $\mathcal{U}$ of $(0,0)$. Suppose that $F(0,0) = F_x(0,0) = F_y(0,0) = F_{xy}(0,0) = 0$ ...

**7**

votes

**1**answer

333 views

### When two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$ are the same?

Let us consider two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$, where $G$ and $G'$ are finite subgroups of $\mathrm{GL}(n,\mathbb{C})$ acting linearly.
It is easy too see, that a different ...

**2**

votes

**1**answer

154 views

### Is there an invariant similar to the delta invariant that distinguishes an $A_2$ node form an $A_1$ node?

Consider the following question: If two nodes collide what do you get?
First of all it can not be a strict $A_2$ node, because the delta invariant
of that is $1$. So it has to be more singular than ...

**2**

votes

**1**answer

281 views

### Thom's gradient conjecture and analyticity

Suppose we have an analytic function $f: U \to {\mathbb R}$, where $U\subset {\mathbb R}^n$ an open subset, with $0\in U$ a critical point of $f$. Thom conjectured that if a trajectory $x(t)$ of ...

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votes

**2**answers

284 views

### Obstructions to being a hyperplane section or a fibre of a Lefschetz pencil

Given a smooth projective variety $X$, when could $X$ fail to be a hyperplane section in some other variety $Y$, or fail to be the fibre of some Lefschetz pencil $\widetilde{Y} \rightarrow ...

**0**

votes

**1**answer

259 views

### What is the simplest way to show that a section of a vector bundle is transverse to the zero set

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

**3**

votes

**2**answers

823 views

### regular singularities and logarithmic singularities

What is the difference between regular singularities and logarithmic singularities? Could someone give me a reference where the distinction is clearly explained? I apologize in advance if this ...

**3**

votes

**1**answer

188 views

### What is the simplest way to represent a $D_5$ singularity?

Consider this curve $f(x,y)=0$ given by
$$ f(x,y) := y^3 + y^2 x + x^4 =0.$$
Is it obvious that after a change of coordinates near the origin, this
curve is equivalent to
$$ \hat{y}^2 \hat{x} + ...

**2**

votes

**1**answer

334 views

### Log resolutions on surfaces and 3-folds in characteristic p

If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the ...

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votes

**3**answers

642 views

### hyperplane sections of isolated hypersurface singularities.

Given an isolated singularity $p$ in a hypersurface $Y$ of dimension $n$ (let say a surface in $\mathbb{P}^3$). I can intersect $Y$ with a hyperplane $H$ passing through $p$ such that it induces a ...

**3**

votes

**1**answer

407 views

### Is it true that the only singularities upto codimension seven are the ADE singularities?

I have a very concrete question about degree $d$ curves in $\mathbb{P}^2$.
Let $$\mathcal{D} \approx \mathbb{P}^{\delta_d}$$
be the space of homogeneous degree $d$ polynomials in three variables upto ...

**3**

votes

**1**answer

203 views

### Is there an upper bound and a lower bound on the contribution to the genus, for a singularity of codimension k?

To make my question precise, suppose you have a complex curve locally given by
$$f(x,y) =0 $$
and $f$ has singularity of type $\chi_k$ at the origin. The codimension of
this singularity is $k$. Let ...

**3**

votes

**3**answers

488 views

### Closure of singular points

Let $f(x,y)$ be a complex degree $d$ polynomial that has this particular
form.
$$ f = \frac{f_{02}}{2} y^2 + \frac{f_{21}}{2} x^2 y +
\frac{f_{12}}{2} x y^2 + \frac{f_{03}}{6} y^3 + ...

**6**

votes

**4**answers

816 views

### I was wondering if the set of singular loops is a (somewhere) submanifold of loop space?

The set of all smooth maps $S^1\to M^n$ ($M$ is a smooth manifold) is a generalized manifold(see http://ncatlab.org/nlab/show/smooth+loop+space).
I was wondering if the set of singular loops (maps ...

**2**

votes

**1**answer

189 views

### Does the Newton polytope characterize the equisingular i.e topological type?

Whenever, people talk about singular plane curves they talk about their Newton polytope which is obviously coordinate dependent. I understand that with some conditions over the singular curve, some ...

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votes

**3**answers

426 views

### Applications for knowing the singularities parametrized by the boundary of a moduli space

Given a moduli space $M$ of some smooth algebraic geometric object such as curves, surfaces, etc. Let $\overline{M}$ be a compactification of $M$. Then, $\overline{M}\setminus M$ introduces ...

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vote

**0**answers

179 views

### When does the smoothing of projectivized tangent cone lift to a deformation of a space?

Let $(X,0)\subset(\mathbb{C}^N,0)$ be the (formal) germ of a singular space (isolated singularity). Let $\mathbb{P}T_{(X,0)}\subset\mathbb{P}^{N-1}$ be its projectivized tangent cone (considered as a ...

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votes

**1**answer

684 views

### the blowing up of a plane curve playing me tricks.

Sorry for the easy question but this is driving me crazy. Consider the blowing up of the curve $(y^2-x^3)^2+y^5$ at the origin.
On the first blowing up, on the chart that intersects the exceptional ...

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votes

**3**answers

2k views

### Wanted: example of a non-algebraic singularity

Given a finitely generated $\def\CC{\mathbb C}\CC$-algebra $R$ and a $\CC$-point (maximal ideal) $p\in Spec(R)$, I define the singularity type of $p\in Spec(R)$ to be the isomorphism class of the ...

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votes

**1**answer

404 views

### Ordinary n-uple Points and Resolution of Singularities on a Surface

Let $X$ be an algebraic variety over some algebraically closed field $\Bbbk$ and let us assume $\dim(X)=2$, i.e. $X$ is an algebraic surface.
First, I would like to know the definition of an ordinary ...

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votes

**1**answer

636 views

### de jong's alteration theorem for families

What is the current status of de Jong's smooth alteration theorem for a family of schemes?
His 1997 paper shows that given any family of curves $X/S$ with $S$ of finite type (and, say, local) over a ...

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votes

**1**answer

364 views

### Resolution of “nice” and zero-dimensional singularities on a surface

Assume I have a singular algebraic surface $X$ over an algebraically closed field (characteristic zero if you want) which is singular in a finite set of points. I am looking for a condition as to the ...

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votes

**2**answers

434 views

### The exceptional locus of a minimal resolution of singularities

Let X be a surface. (A surface is an excellent integral normal separated 2-dimensional scheme.)
Let $\psi:Y\longrightarrow X$ be a minimal resolution of singularities and let $E$ be an irreducible ...

**2**

votes

**1**answer

296 views

### Stable singularities of smooth map $\mathbb R^3\to \mathbb R^4$

Does anybody know any classification of stable singularities of smooth map $f:\mathbb R^3\to \mathbb R^4$?
It is clear that there are singularities which look like intersection of 2 (or 3 or 4) ...

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votes

**1**answer

967 views

### Du Val singularity of type G=A,D,E and “small” representations of G

We all know that a simple singularity $W_G(x_1,x_2,x_3)=0$ of type G=A,D,E has the following nice deformation involving the Cartan subalgebra $\mathfrak{h}$ of the Lie algebra $\mathfrak{g}$ of $G$. ...

**2**

votes

**0**answers

135 views

### Tangent cones to Severi strata

Let $\mathbb{C}[[x,y]]/f(x,y)$ be a reduced plane curve singularity. The base of a versal family can be taken to be (an open subset in) $\Lambda = \mathbb{C}[x,y]/(f,\partial_x f, \partial_y f)$; the ...

**1**

vote

**1**answer

397 views

### Brieskorn's proof of a theorem by Milnor about the Milnor number

I am looking for a reference or short explanation of a proof by E. Brieskorn.
In his famous work "Singularities of complex hypersurfaces" Milnor proves that the (nowadays called) Milnor Number (in ...

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votes

**1**answer

232 views

### Transversals to singular subvarieties

Say $\mathbb{C}^d \subset Y^{N-k} \subset \mathbb{C}^N$ are closed imbeddings of complex analytic subvarieties of the indicated dimension, $Y$ is not smooth. At a point $y \in Y$, a generic, ...

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votes

**2**answers

478 views

### Stratified Pseudomanifold

Hi there,
I have a, I guess, simple question.
In the definition of an n-dimensional stratified pseudomanifold one demands the following filtration
$X=X_n \supset X_{n-1}=X_{n-2} \supset X_{n-3}\supset ...

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votes

**4**answers

458 views

### Smooth in codimension-k and the weight filtration

Let $X$ be an algebraic variety. Then $H_{et}^k(X)$ has a filtration whose associated graded pieces are labeled by "weights", certain integers between $0$ and $2k$. If $X$ is smooth, then the weights ...

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votes

**2**answers

3k views

### Elementary short exact sequence of sheaves

This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow ...

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vote

**2**answers

345 views

### Can one obtain surfaces with interesting invariants as resolutions of singular surfaces?

(Perhaps a not very well defined question)
Let $(S_t)_t$ be a (flat) family of compact complex surfaces. Assume the generic member is smooth while $S_0$ has isolated singularities. As the simplest ...

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votes

**0**answers

457 views

### An elementary question in singularities

The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ...

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votes

**4**answers

1k views

### minimal resolution of singularities

What is the minimal resolution of singularities of the surface
$S^2(X^3+Y^3+Z^3)-3(S^2+T^2)XYZ=0$ which is a subset of $\mathbb{P}^1\times\mathbb{P}^2$
Please note that in this equation ...

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votes

**0**answers

2k views

### Microlocal geometry - A theorem of Verdier

(1) In "Geometrie Microlocale", Verdier states the following theorem.
Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$.
Then for $\ell$ a linear form on $E$, we have a ...

**2**

votes

**3**answers

338 views

### Alterations factor as modification + finite map

I'm learning about de Jong's theory of resolution of singularities and the following fact is used numerous times: an alteration of varieties $h: X \rightarrow Y$ factors as $X \xrightarrow{\pi} Z ...

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votes

**0**answers

256 views

### Singularity structure of integrals of rational functions

Suppose I have a convergent integral of the form $\int_0^1dx_1\dots\int_0^1 dx_n \frac{P(x_i)}{Q(x_i)}$, where $P$ and $Q$ are polynomial functions of $n$ nonnegative real variables $x_i$. Let the ...

**2**

votes

**2**answers

405 views

### on the relative conductor of curve singularity and quotient of ideals

Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ ...

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vote

**2**answers

850 views

### crepant resolution

Let's put $m>n$ two nonnegative integers and $Gr:=Grass(n,k^m)$ the grassmanian of the subspaces of dimension $n$ in $k^m$. We have a natural immersion $Gr \subset P({\Lambda}^{n} k^m)$ and I call ...