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4
votes
1answer
339 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 ...
2
votes
1answer
559 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 ...
2
votes
1answer
351 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 ...
2
votes
2answers
386 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
1answer
262 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) ...
9
votes
1answer
951 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
0answers
118 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
1answer
375 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 ...
4
votes
1answer
227 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, ...
3
votes
2answers
416 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 ...
5
votes
4answers
449 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 ...
13
votes
2answers
2k 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 ...
1
vote
2answers
343 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 ...
6
votes
0answers
447 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 ...
5
votes
4answers
954 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 ...
26
votes
0answers
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
3answers
307 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 ...
6
votes
0answers
241 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
2answers
384 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 $R\subset R'\subset \bar{R}$ ...
1
vote
2answers
799 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 ...
7
votes
1answer
419 views

Simplified treatment of resolutions of complex analytic varieties?

According to the article of Hauser: The Hironaka theorem on resolution of singularities http://www.ams.org/journals/bull/2003-40-03/S0273-0979-03-00982-0/home.html The existence of resolution of ...
12
votes
5answers
672 views

Comparing fundamental groups of a complex orbifolds and their resolutions.

Let $X$ be a complex manifold with quotient singularities, and let $\tilde X$ be its resolution (that exists, for example, by Hironaka). Then I am pretty sure that $\pi_1(X)\cong \pi_1(\tilde X)$. ...
12
votes
4answers
977 views

What formal properties should resolution of singularities have?

If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...
11
votes
4answers
974 views

Singular semi-Riemannian Geometry: usefulness and state of the art

My question has two parts, one concerning the state of the art of the subject, and the other the usefulness. 1. State of the art. Can someone provide references reflecting the state of the art in ...
23
votes
7answers
3k views

Examples of Mixed Hodge Structures

Does anyone know a user-friendly, example-laden introduction to mixed Hodge structures? I get from Wikipedia how to calculate for a punctured and pinched curve ...
0
votes
0answers
92 views

Singularity links of quotients

Hi there, I have a question concerning the quotient $\mathbb{C}^3/\mu_p$ where $\mu_p=\{\zeta\in \mathbb{C}^3|\zeta^p=1\}$ acts on $\mathbb{C}^3$ via $\zeta(x,y,z)=(\zeta^{-1}x,\zeta^{-p}y,\zeta z)$. ...
2
votes
2answers
258 views

How can I compute the full set of nodes of a surface?

The nodes of a surface are special cases of more general singularities. For example, the Cayley cubic has four nodes. The full set of singularities of a surface can be characterized by finding all ...
6
votes
2answers
1k views

Resolution of singularities

What is the relation between crepant resolutions and minimal resolutions? Are they the same thing?
2
votes
2answers
832 views

Explicit examples of resolution of (projective) 3-folds over k?

I'm looking for examples of explicit resolutions of (projective) 3-folds over a field k (char 0), with isolated singularities, or at least with smooth singular locus. I've looked in various books and ...
8
votes
3answers
1k views

Singularities of pairs

In the next days I have to give a talk in which I need to explain some of the usual singularities of pairs that one meets when dealing with the minimal model program: KLT, DLT and LC pairs. In ...
7
votes
1answer
657 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
386 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
459 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
379 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
265 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 ...
3
votes
2answers
1k 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
402 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
290 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
302 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
566 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
474 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 ...
8
votes
2answers
487 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
691 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
437 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
301 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
555 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
411 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
861 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
477 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
1k 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? ...