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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 ...
10
votes
4answers
941 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 ...
12
votes
5answers
655 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 $\pi_1(X)\cong \pi_1(\tilde X)$. ...
10
votes
2answers
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
1answer
470 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. ...
2
votes
2answers
267 views

Does a generic curve inside the space of curves with a node at a specific point have only finitely many nodes?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$ polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$. Define $\mathcal{A}$ to be ...
1
vote
1answer
184 views

smooth quotient out of a singular variety?

If $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $G$ is a finite group acting faithfully on $X$, then the Shepard-Todd theorem gives us some criterion for $X/G$ to be smooth. My ...
3
votes
1answer
251 views

Normal form for a holomorphic Morse function

Similarly to Morse lemma, a holomorphic Morse function can be written, near a critical point, as $W_1^2+W_2^2+...+W_n^2+C$ , for well chosen coordinates $W_1,W_2,...,W_n$. I want to cite this result ...
3
votes
0answers
354 views

Isolated singularities and tangent cones

Assume that I have an affine hypersurface $X =V(f)\subset \mathbb{C}^4$ of degree $d$ with an isolated singularity of multiplicity $m$ at the origin $o=(0,0,0,0)$. Let $$f:=f_m + f_{m+1}+ \cdots ...
7
votes
2answers
376 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 ...
6
votes
2answers
611 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 $F(0,0) = F_x(0,0) = F_y(0,0) = F_{xy}(0,0) = 0$ ...
8
votes
2answers
546 views

Whitney stratifications

Many results on characteristic classes of singular varieties (as well as other singularity-theoretic constructions) make use of a so-called "Whitney stratification" of the variety under consideration, ...