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6
votes
1answer
631 views

Rational singularities for fibered surfaces

This question consists of two parts. I will try to be as short and clear as possible. Let $S$ be a Dedekind scheme of characteristic zero. The main examples are $\mathbf{P}^1_k$, with $k$ a field of ...
5
votes
2answers
376 views

Tame ramification of (mild) curve singularities.

Suppose that $C$ and $D$ are curves of finite type over an algebraically closed field $k$ (we make some of these hypotheses for simplicity). We view these as pointed curves with singularities $c \in ...
2
votes
2answers
445 views

Higher dimensional nodes

A node on a curve is a singular point that locally looks like the intersection of two lines. I think the precise way to say this is that $p \in X$ is a (closed?) point on a scheme $X$ (of finite type ...
1
vote
1answer
357 views

complex singularity exponent, lct

Hi everybody, I have a question about log canonical thresholds / complex singularity exponents. If I understood well, this invariant sees more things than the multiplicity, for example, the cusp in ...
2
votes
1answer
264 views

Is resolution of singularities effective?

Suppose I have a singular projective variety defined by some homogeneous equations in complex projective space. Is the resolution of singularities effective? That is, do I actually know which smooth ...
2
votes
2answers
953 views

Are orbifold singularities canonical?

This is a direct consequence of my previous question: Extending group actions on varieties In his answer, inkspot said that group actions can be extended if the variety has ample canonical class and ...
0
votes
1answer
387 views

Does any one understand the details of M Kazarian's work in enumerative geometry of $\mathbb{C}\mathbb{P}^2$ ?

I wanted to know if anyone understood the details of the paper "Multisingularities, cobordisms, and enumerative geometry" available at the site http://www.mi.ras.ru/~kazarian/. In particular does ...
8
votes
0answers
273 views

Which presentations of (non)planar algebras give rise to knots?

Reidermeister's theorem states that the set of knots, modulo ambient isotopy, is isomorphic to the planar algebra generated by crossings, modulo Reidemeister moves. This planar algebra presentation is ...
4
votes
1answer
300 views

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 ...
1
vote
2answers
561 views

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 ...
4
votes
3answers
459 views

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 ...
7
votes
2answers
478 views

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 ...
4
votes
2answers
677 views

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 ...
6
votes
3answers
430 views

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 ...
0
votes
1answer
289 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 $b(|L|)$. ...
6
votes
2answers
522 views

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 ...
3
votes
3answers
400 views

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 ...
6
votes
3answers
835 views

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 ...
4
votes
1answer
449 views

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, ...
3
votes
2answers
988 views

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? ...
3
votes
4answers
859 views

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 ...
6
votes
3answers
633 views

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 ...
18
votes
5answers
1k views

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 ...
6
votes
1answer
677 views

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 $\lambda \in A^1$ ...
26
votes
2answers
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 ...
2
votes
1answer
187 views

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 ...
5
votes
1answer
567 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 ...
2
votes
1answer
486 views

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}, ...
3
votes
0answers
396 views

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 ...
4
votes
1answer
402 views

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 ...