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255 views

Explicit Computations of Dynkin Diagrams of Isolated Singularities

Let $f$ be a complex polynomial with an isolated singularity at the origin. Take a Morse deformation $\tilde{f}$, and consider the braid group action on the set of distinguished bases of vanishing ...
3
votes
0answers
200 views

Good covers on complex algebraic varieties with normal crossings singularities

Let $X$ be a topological space. A good cover on $X$ is an open cover such that all finite non-empty intersections are contractible. It is a theorem of Hironaka that (complex) algebraic sets admit ...
1
vote
1answer
344 views

triple point singularity

Assume a complex surface $X$ admits a fibration structure over $\mathbb{CP}^1$ with some singular fiberes. Are there explicit examples of such surfaces with triple point singularity?
3
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1answer
302 views

$A_{\infty}$ singularity

What kind of singularity is commonly meant by $A_{\infty}$?
35
votes
2answers
2k views

When is a singular point of a variety smooth?

If $X$ is a nonsingular algebraic (or analytic) variety over $\mathbb C$ or $\mathbb R$ then it is certainly $C^\infty$ over the reals. The converse is false for a silly reason : in the real or ...
5
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0answers
288 views

What is known about “singularity types” in the Murphy's Law sense?

In his "Murphy's Law" paper, Vakil gives a definition equivalent to the following: The singularity type of a pointed scheme $(X,p)$ its equivalence class, under the following equivalence relation: ...
9
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3answers
505 views

Singular fibers of generic smooth maps of negative codimension

This is in some sense a follow-up to my question on submersions. Let $f\colon\thinspace M\to N$ be a generic smooth map between closed manifolds of dimensions $m$ and $n$. Assume that the codimension ...
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vote
0answers
188 views

Do deformations of isolated hypersurface singularity naturally induce deformations of their divisors?

Let $0 \in V =(f=0) \subset \mathbb{C}^{n+1}$ be an affine variety with an isolated hypersurface singularity at the origin for $n \ge 3$. Let $0 \in D=(x=f=0) \subset V$ be a divisor with only ...
4
votes
0answers
202 views

Fixed point sets that carry topology

Let $M$ be a closed smooth manifold. A generic diffeomorphism $\phi: M\rightarrow M$ has non-degenerate fixed points, i.e. the intersections of its graph with the diagonal in $M\times M$ are all ...
2
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0answers
139 views

Controlling singularities on log mmp

Suppose all my varieties are complex threefolds $X\rightarrow Y$ over some smooth base curve germ $Y$. We can assume the fibres are Del Pezzo surfaces with generic smooth fibre. If I do (relative) ...
3
votes
1answer
501 views

Is P^2 important in Kontsevich's recursion formula?

There is a famous recursion formula by Kontsevich to find the number of genus zero degree $d$ curves in $\mathbb{CP}^2$ through $3d-1$ points. My question is the following: Let $S$ be a complex ...
4
votes
2answers
524 views

Q-factorial and rational singularities on surfaces

Let $X$ be a normal surface. Is any rational singularity $\mathbf{Q}$-factorial? I've seen this somewhere for surfaces over fields, but what about the general case of an integral 2-dimensional ...
2
votes
0answers
144 views

Are there ways to make low degree checks for enumerative formulas except for curves in CP^2?

This is a concrete question in Enumerative geometry. Let $S$ be a compact complex surface and $L\rightarrow S$ a holomorphic line bundle. Let $$ \delta_d = \text{dim}~ \mathbb{P}(H^0(S,L^d)) $$ ...
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1answer
400 views

Can a PDE constrain the degree of a $C^\infty$ map germ?

Let $\pi:E\to M$ be a smooth vector bundle over a smooth manifold, with $\text{rank}(E)=\text{dim}(M)$. For a section $\sigma$ of $E$ with a zero at $p\in M$, define the degree of the zero at $p$ to ...
4
votes
2answers
575 views

Can one prove vanishing of higher direct images fiber-wise?

Let $\pi:X\to Y$ be a proper map of algebraic varieties (over $\mathbb C$) which is a bi-rational equivalence. are the following statements equivalent? The derived direct image of $O_X$ is $O_Y$. ...
1
vote
1answer
165 views

How can one bound 'the lower perverse degree' for a constant sheaf on a variety that is smooth in high codimension?

Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of ...
7
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0answers
436 views

Physicists Euler number conjecture

Physicist's Euler number conjecture says: If $G \subset SL(n,\mathbb{C})$ is a finite group, $X=\mathbb{C}^n/G$ is the quotient space and $f:Y \rightarrow X$ a crepant resolution (always exists for ...
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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
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1answer
482 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. ...
7
votes
2answers
386 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 ...
5
votes
2answers
1k views

construct the elliptic fibration of elliptic k3 surface

Hi all, As we know, every elliptic k3 surface admits an elliptic fibration over $P^1$, but generally how do we construct this fibration? For example, how to get such a fibration for Fermat quartic? ...
4
votes
1answer
288 views

Property of singularity

Let $X$ be an algebraic variety, $S \subset X$ its singular locus, and $x \in S$. Say that $x$ is good if for any neighborhood $U$ of $x$, any top differential form $\omega$ on $U \setminus S$ and ...
4
votes
2answers
549 views

Vanishing associated to a resolution of singularities

Let $\pi: V\to W$ be a resolution of singularities, let $E \subset V$ be the exceptional divisor, and let $F$ be a coherent sheaf such that $R^i\pi_*F=0$ for $i>0$. Can we conclude that ...
5
votes
2answers
836 views

Places to learn about Landau-Ginzburg models

Here is what I know about Landau-Ginzburg models: It is an important player in the story of mirror symmetry. It involves "potentials" which are functions of complex varibles, which have isolated ...
6
votes
2answers
395 views

How can we find a surface with a given singularity?

I was surprised the first time I learned that a quintic plane curve can have an $A_{10}$ singularity i.e $x^2+y^{10}$. I am wondering if there is something about that phenomenon: Given a singularity ...
7
votes
2answers
619 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
314 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
139 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
256 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
272 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
244 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
660 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
176 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
282 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
591 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
374 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
195 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
481 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
802 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
173 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 ...
8
votes
3answers
389 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
171 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
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1answer
499 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 ...
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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
332 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
552 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
341 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
372 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
252 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
947 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 $G$. ...