The singularity-theory tag has no wiki summary.

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### How to enumerate curves with a singularity once you know the corresponding Thom Polynomial?

Does any one know how to go from "Thom polynomials" to "Enumerating curves". I believe there
is a relation between the two, but I don't know how to go about it. Let me make the
question more ...

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**3**answers

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### Poll about your proof of resolution of singularities and a request for advice

The questions first: What is the proof of resolution of singularities that you know?
Why am I asking?: There are a number of proofs of resolution of singularities of varieties over a field of ...

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**3**answers

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### A necessary and sufficient condition for a curve to have an $A_k$ singularity.

Hi
Does any one know of a necessary and sufficient condition for a curve to have a singularity
of type A_k.
More precisely, a curve f=0 has a singularity of type A_k at a point, if there exist
local ...

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votes

**2**answers

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### Small neighborhoods of singularities on varieties

In Singular points of complex hypersurfaces, John Milnor proves the following theorem:
Let $x \in V$ be a point on a variety $V$ in $\mathbb{R}^n$ or $\mathbb{C}^n$. Assume $x$ is either a smooth ...

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votes

**2**answers

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### How/where are semi-log resolutions used?

In the paper, by János Kollár there is problem 19 (page 8).
It is one more strict resolution. A resolution that leaves untouched the semi-simple-normal-crossings singularities of pairs.
My question ...

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votes

**3**answers

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### What strict resolutions of singularities are needed?

Suppose we have a collection, $S$, of singularities types and consider a resolution of singularities (this is: a proper birrational morphism $Y\rightarrow X$ such that Y only contains singularities of ...

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votes

**1**answer

302 views

### Log resolutions of linear series

Let $X$ be a complex normal projective variety, let $|L|$ be a non empty linear series on $X$ and let $b(|L|)$ be its base ideal.
Suppose $f:X'\rightarrow X$ is a log resolution of the ideal $b(|L|)$.
...

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votes

**2**answers

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### Is the desingularization of a normal variety with only quotient singularities projective

The base field will be the field of complex numbers. I have a slightly technical problem concerning the resolution of singularities of a certain variety. Basically, I want to to know if it is ...

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**3**answers

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### The link of a singular quintic hypersurface in CP^4

Given a family of quintic hypersurfaces in $\mathbb{CP}^4$ by
$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5+(5+\epsilon)x_1x_2x_3x_4x_5$
we get a singular variety for $\epsilon=0$ with 125 singular points.
I know ...

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**3**answers

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### Is there an obvious way for showing singularities are quotient?

I'm stuck on a technicality concerning singularities.
Basically, I have to show that the singularities of a $\mathbf{certain}$ normal projective variety over $\mathbf{C}$ are rational. (I won't ...

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votes

**1**answer

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### When singular points of a reduced scheme are not dense in it?

A stupid AG question: could singular (Zarisky) points be dense in a reduced (Noetherian) scheme $S$? If yes, which 'standard' restrictions on $S$ could ensure that this does not happen? For example, ...

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**2**answers

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### On minimal resolution of singularities and the type of singularities

Let $Y$ be a normal projective surface, let $X$ be a smooth projective surface and let $\pi:Y\longrightarrow X$ be a finite morphism. Why are all singularities of $Y$ cyclic quotient singularities? ...

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votes

**4**answers

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### Matrix factorization categories for ADE singularities

What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated.
Background: For ADE singularities, see for example this. For ...

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votes

**3**answers

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### What does being Analytically Isomorphic imply for classification of singularities on curves?

Hartshorne I.5 mentions the definition of being analytically isomorphic:
P on X and Q on Y are analytically isomorphic iff the completion of O_P is isomorphic to the completion of O_Q where the ...

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**5**answers

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### Is a 'generic' variety nonsingular? Or singular?

I'd like to know whether there's some coherent meaning of 'generic' for which one can say that a 'generic' variety over an algebraically closed field $K$, say, is nonsingular or singular. We could ...

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**1**answer

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### Localization of vanishing cycles

Consider a regular holonomic D-module (or a perverse sheaf) $M$ on a smooth variety $X$. Let $f:X\to A^1$ be a polynomial (or holomorphic) function.
Question: Is it true that the $\lambda \in A^1$ ...

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**2**answers

1k views

### Limit of a series of singularities

The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities ...

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**1**answer

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### Existence of smoothing of Calabi-Yau cones over $dP_{1}$ and $dP_{2}$

The blowdown of the zero section of the canonical bundle of the first del Pezzo surface $dP_{1}$, the blowup of $CP^{2}$ at one point, is a Calabi-Yau cone. I was just wondering if this cone admitted ...

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votes

**1**answer

594 views

### Resolution of Singularities, Nature of

Hironaka's theorem guarantees an existence of resolution of singularities in characteristic 0. If I am not wrong, it also guarantees (or at least some other result does), that if the resolution is a ...

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votes

**1**answer

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### Corank 4 hypersurface singularities

A function f: ($\mathbb{C}^n$,0) $\to$ ($\mathbb{C}$,0) is considered a hypersurface singularity if the point $(0,0,\dots,0)$ is the only point in the ideal $\langle \frac{\partial f}{\partial x_1}, ...

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### Sebastiani-Thom isomorphism for D-modules

Considering $f:X\to \mathbb{C}$, $g:X\to \mathbb{C}$ and $f\oplus g:(x,y)\mapsto f(x)+g(y)$.
The Sebastiani-Thom isomorphism is an isomorphism $\Phi_{f\oplus g}(M\boxtimes N) = \Phi_{f}(M) \otimes ...

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**1**answer

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### Singularity theory references

I am looking for some good references on singularity theory. I'm interested in singularity theory in the context of mirror symmetry, so this means I'm interested in things like Picard-Lefschetz ...