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6
votes
2answers
567 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
1answer
298 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
1answer
136 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
1answer
239 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 ...
3
votes
2answers
263 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
1answer
231 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
2answers
599 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
1answer
173 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
1answer
268 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 ...
6
votes
3answers
566 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
1answer
359 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
1answer
190 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
3answers
477 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
4answers
789 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
1answer
169 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 ...
7
votes
2answers
313 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 ...
1
vote
0answers
159 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 ...
3
votes
1answer
471 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 ...
26
votes
3answers
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 ...
4
votes
1answer
307 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
520 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
336 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
350 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
234 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
942 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
113 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
350 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
226 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
385 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
441 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 ...
12
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
341 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
435 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
880 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 ...
25
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
285 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
238 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 ...
1
vote
2answers
355 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
740 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
398 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
635 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
962 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 ...
10
votes
4answers
912 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 ...
22
votes
7answers
2k 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
255 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
793 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 ...
6
votes
1answer
630 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 ...