The singularity-theory tag has no wiki summary.

**1**

vote

**0**answers

70 views

### How would you call a variety that is locally a complete intersection up to defect c?

Let $X$ be an equidimensional variety of dimension $n$ over a field that can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in $P^N$ (for a large enough $N$; we ...

**0**

votes

**1**answer

147 views

### Number of singular fibers in families of hypersurfaces

Consider the projection map
$$\pi: X = V(t_0 f + t_1 gh) \to \mathbf P^1,$$
where $[t_0: t_1]$ are the homogeneous coordinates on $\mathbf P^1$, $f=f(x_0, \dots, x_n)$ is a homogeneous polynomial of ...

**0**

votes

**0**answers

104 views

### When can one find holomorphic sections vanishing at a point to a certain order?

Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements:
Statement $A_0$: Given any point $p\in X$, there ...

**0**

votes

**1**answer

159 views

### A condition on isolated singularity

Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = ...

**4**

votes

**0**answers

92 views

### Real structure in the mixed Hodge structure associated to an isolated singularity

We know that a mixed Hodge structure on a complex vector space $H$ with an integral lattice $H_{\mathbb Z}$ consists of the weight filtration and the Hodge filtration. For an isolated hypersurface ...

**3**

votes

**1**answer

172 views

### Obtaining non-normal varieties by pushout

In his answer to this MO question, Karl Schwede claimed that every non-normal variety can be obtained by an appropriate pushout diagram, as sketched in that answer. This would give substance to the ...

**0**

votes

**0**answers

43 views

### Conifold singularities on the mirror quintic

It is well-known that conifold transitions go in the reversed direction under mirror symmetry. My question is, what is the mirror picture of a degeneration of the generic quintic CY3 to the one ...

**3**

votes

**2**answers

234 views

### How can one determine if a singularity is simple?

Let $f(z_1,z_2,\dots ,z_n)$ be an analytic function in $\mathbb{C}[[z_1,z_2,\dots ,z_n]]$ whose leading term defines an isolated singularity at the origin.
If we have the following types of ...

**0**

votes

**1**answer

158 views

### Discriminant of a polynomial in two variables

I want to compute the discriminant of the following polynomial
$$
F(X,Y)=X^mY^n+\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}c_{ij}X^iY^j.
$$
Here the discriminate means the equation $D(c_{i,j})$ in the variables ...

**3**

votes

**1**answer

79 views

### Singularities induced by the toric ambient spaces

Let $\Delta \subset \mathbb{R}^4$ be a (reflexive) polytope and $X$ be the hypersurfacedefined by a generic section of the any-canonical bundle of the toric variety $\mathbb{P}_{\Delta}$. Are there ...

**0**

votes

**0**answers

67 views

### on lifting elements in a tangent space

Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$
Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and ...

**4**

votes

**0**answers

166 views

### $n$-Fold Framed Functions

Suppose that $M$ is a manifold. One can consider a suitably constructed space of generalized framed Morse functions on $M$, let's call it $\mathrm{Fun}^\mathrm{fr}(M)$. This space is known to be ...

**13**

votes

**1**answer

337 views

### Higher Cerf Theory

Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that ...

**0**

votes

**0**answers

114 views

### A question on a toric singularity

Recently I was told that the threefold singularity
$$
x^2+y^2+z^2+w^{2n}=0
$$
in $\mathbb{C}^4$ is a toric variety. How can I see this? What are the generators of the toric fan? Moreover, is it true ...

**6**

votes

**3**answers

227 views

### Contracting a rational curve in a Calabi-Yau threefold

Let $X$ be a Calabi-Yau threefold and $C \subset X$ be a rational curve with $N_{C/X}\cong \mathcal{O}\oplus \mathcal{O}(-2)$. Can one contract the curve $C$? Assuming the answer is yes, what kind of ...

**7**

votes

**0**answers

279 views

### Singularities arising from the Minimal Model Program (an algebraic point of view)

I will start the story by the end:
Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ...

**5**

votes

**2**answers

298 views

### Singular points of algebraic varieties and parametrization by Puiseux series

Let $V\subset \mathbb{R}^n$ (or $\mathbb{C}^n$ if that makes anything easier) be an algebraic variety and $p\in V$ a possibly singular point. Let $U\subset V$ be a sufficiently small neighborhood of ...

**1**

vote

**0**answers

51 views

### on reductive monoids which are gorenstein

Let $M$ a reductive monoid, i.e. a integral normal affine scheme, which is a monoid whose group of units is a connected reductive group.
By Rittatore ...

**1**

vote

**0**answers

52 views

### Complexity of mappings (forms) in R. Thom's “Structural stability and morphogenesis”

In his "Structural stability and morphogenesis", R. Thom (especially in the chapter about dynamics of forms) among other things speculates about a notion of complexity of a "form" (mapping between ...

**2**

votes

**0**answers

116 views

### generalization Abhyankar's lemma

This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question.
Let ...

**2**

votes

**1**answer

242 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 ...

**6**

votes

**1**answer

254 views

### Picard group generated by effective divisors: counterexample?

Let $X$ be an integral variety defined over an algebraically closed field $k$ of characteristic 0 with finitely generated Picard group $Pic(X)$ and such that $k[X]^\times=k^\times$ (i.e. the only ...

**5**

votes

**1**answer

279 views

### A geometric characterization of smooth points of a complex algebraic variety

Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric.
Fix an arbitrary point $p\in X$. Let ...

**4**

votes

**1**answer

130 views

### Castelnuovo's rationality criterion on singular surfaces?

Let $S$ be a projective surface over an algebraically closed field. Suppose that $q(S)=h^1(\mathcal O_S)=0$ and $P_2(S)=h^0(\mathcal O_S(2K_S))=0$. If $S$ is smooth, Castelnuovo's rationality ...

**2**

votes

**0**answers

131 views

### A strong form of implicit function theorem (what happens when the derivative is degenerate?)

(this can be considered as some ad)
Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...

**0**

votes

**0**answers

72 views

### Depth of conormal sheaf of a quotient singularity in a smooth variety

Let $U= \mathbb{C}^n/ \mathbb{Z}_r$ be a cyclic quotient singularity by the finite cyclic group $\mathbb{Z}_r$ of order $r$.
Assume that the group action is free outside a closed subset $Z \subset ...

**4**

votes

**2**answers

254 views

### Varieties with big anti-canonical divisor

I recently heard about the following problem:
Let $X$ be a projective variety with klt singularities and such that $-K_X$ is big. Is $X$ a Mori Dream Space ?
Now, $-K_X$ big if and only if $-K_X ...

**1**

vote

**1**answer

97 views

### Complement of bifurcation variety

I am reading a seminal paper of Arnold "Normal forms of functions near degenerate critical points, the Weyl group of $A_k$, $D_k$, $E_k$ and lagrangian singularities".
Let $f\colon \mathbb{C}^n\to ...

**0**

votes

**0**answers

79 views

### Birational contraction to a $\mathbb{Q}$-Gorenstein Variety

Given a birational contraction morphism $X\rightarrow Y$
of complex normal algebraic varieties.
If $Y$ is a smooth variety, what kind of singularities can appear
on $X$?
I would be grateful of any ...

**1**

vote

**1**answer

138 views

### Intrinsically proving a singularity is rational

In general, how to prove a variety has rational singularities intrinsically? i.e., don't use the Artin's criterion concerning the exceptional locus. And what kinds of varieties have only rational ...

**5**

votes

**1**answer

217 views

### The singularity of the algebraic stack and the singularity of the coarse moduli space

It is possible that an algebraic stack is smooth while the coarse moduli space is not smooth. I want to know what is relationship between the singularity of the algebraic stack and that of its coarse ...

**3**

votes

**1**answer

147 views

### Simple example of isolated critical point with non-semisimple monodromy

Consider a polynomial map $f :\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ with $f(0)=0$ (no constant term) and with isolated critical point at $0 \in \mathbb{C}^{n+1}$. We can choose a disc $D$ of some ...

**2**

votes

**1**answer

410 views

### Topological degree and polynomial degree

Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ...

**1**

vote

**1**answer

176 views

### Analytic vector fields on surfaces which have infinite number of singularities

Let $X$ be an analytic vector field on a compact oriantable surface $S$ with volume form $\omega$. We denote the set of its singularities by $Z(X)$.
A local question
Is there an analytic vector ...

**1**

vote

**0**answers

309 views

### Presence of singular points in the trajectory of a double pendulum

Watching the trajectory of a double pendulum, I caught myself wondering if it would be possible to prove that the path the second pendulum makes contains "cusps" or singular points. Upon investigating ...

**1**

vote

**0**answers

66 views

### Three- dimensional astroid and catastrophe maps

Is a three-dimensional astroid curve $(2\cos^3u,2\sin^3u,3\cos2u)$ a part of a bifurcation set of some catastrophe map?

**6**

votes

**1**answer

512 views

### Bertini's Theorem

Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...

**6**

votes

**1**answer

98 views

### Properties of singularities that are preserved by categorical quotients

Let $G$ be a reductive group acting on an affine singular
variety $X$, and let $X/G$ be the categorical quotient. I know that if
$X$ has rational singularities, then so does $X/G$ ...

**1**

vote

**0**answers

241 views

### Blowdown and contraction

I am sorry, my question is very naive.
2nd Edit: Let us suppose that $V$ is a smooth complex projective variety, and $Y\subset V$ is a smooth divisor and has an ample conormal line bundle.
We would ...

**2**

votes

**1**answer

210 views

### fearful of defining equivalent germs for non isolated singularities

Two power series $G(x_1, \ldots, x_n)$ and $F(x_1, \ldots, x_n)$ are equivalent over $\mathbb{C}$ if there is an automorphism of the ring
$\mathbb{C}[[x_1, \ldots, x_n]]$ given by $x_1 \to \phi(x_1, ...

**0**

votes

**0**answers

31 views

### Discriminant set and Bifurcation set

the discriminant set of an n-dimensional versal unfolding of
an $A_k$ singularity, and the bifurcation set of an n -dimensional versal unfolding of an $A_{k+1}$, (potential type) singularity are ...

**1**

vote

**1**answer

69 views

### Determining the desingularization from the complete local ring

Suppose I have a curve $C$ over a field $k$ and that $p$ is a singular point of $C$. Let $f : X \to C$ be the desingularization of $C$ at $p$. Then for each $s \in f^{-1}(p)$ we have a map of local ...

**2**

votes

**0**answers

125 views

### geometric irregularities in pde's

The following question is intended for a person more acquainted with the works of Yves Laurent.
see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French)
...

**3**

votes

**2**answers

245 views

### Cohomology of the tangent sheaf of $\mathbb{P}(1,2,3)$

Using the exact sequence
$$0\mapsto\mathcal{O}_{\mathbb{P}^{2}}\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(1)^{\oplus 3}\rightarrow T_{\mathbb{P}^{2}}\mapsto 0$$
it is easy to compute ...

**3**

votes

**1**answer

265 views

### For what varieties do we have results on the category of singularities?

Let $X$ be a singular variety. Define the (triangulated) category of singularities (as in Orlov's paper)
as the Verdier quotient of the derived category of coherent sheaves on $X$ modulo the full ...

**3**

votes

**1**answer

171 views

### Deformations of quotient singularities

Let $Y$ be an affine scheme over a field of characteristic zero. Suppose we have a group $G$ acting on $Y$ and that the subset of $Y$ of points with non-trivial stabilizer is in codimension greater or ...

**5**

votes

**1**answer

219 views

### Relation between Milnor ring and middle dimensional homology of hypersurface

I have suspected that the following is well-known:
If $P$ is a homogeneous polynomial of degree $d$ in $n$ variables (for example, Fermat quintic $x_1^5 + \cdots + x_5^5$). The Milnor ring is ...

**2**

votes

**0**answers

76 views

### Lagrangean equations for the generating function of quadrangulations

Let $M(z)$ be the generating function of edge-rooted connected quadrangulations, with $z$ marking the number of edges. I derived the following Lagrangean equations for $M(z)$:
$$M(z) = ...

**8**

votes

**1**answer

370 views

### Why can you deform singularities in two dimensions but not in higher dimensions?

I've been trying to read this paper to understand deformations of surface quotient singularities. I'm particularly interested in when one can deform certain cyclic quotient singularities into other ...

**6**

votes

**2**answers

261 views

### Whitney stratification and affine grassmanian

Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by ...