The singularity-theory tag has no usage guidance.

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### 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 ...

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### 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 ...

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76 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 ...

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### $n$-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 ...

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474 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 ...

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384 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 ...

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347 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 ...

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### 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 ...

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### 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 ...

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222 views

### Are codimension one foliations of $\mathbb{R}^{n}-\{0\}$ with compact leaves, stable at origin?

Assume that we have a codimension one foliation of $\mathbb{R}^{n}-\{0\}$ with compact leaves.
Is it true to say that the foliation is stable at origin:That is: for every neighborhood $V$ of ...

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### 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 ...

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241 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 ...

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328 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 ...

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292 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 ...

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320 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 ...

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160 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 ...

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229 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/$C^\infty$ ...

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385 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 ...

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101 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 ...

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122 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 ...

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154 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 ...

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381 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 ...

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### 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 ...

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### 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 $L$ ...

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203 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 ...

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348 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 ...

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70 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?

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553 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 $p_1,...,p_n$. ...

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### 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 $X/G$ ...

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361 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 ...

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### 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, ...

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### 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 ...

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### 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)
...

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364 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 ...

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297 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 ...

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221 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 ...

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439 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 ...

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### 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) = ...

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429 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 ...

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### 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 ...

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### Contractibility of curves and embedding into projective space

Let $f:X \to Y$ be a proper surjective morphism of projective surfaces such that there exists a curve $C \subset X$ for which $f|_{X\backslash C}$ is an isomorphism and $f(C)$ is a set of points. ...

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### References for resolutions of ordinary singular points

Let $X$ be a $n$-dimensional complex projective algebraic variety, let us suppose that $X$ has only isolated singularities.
Edit: Let us say that an ordinary $m$-ple singular point is an isolated ...

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### Homology basis of minimal resolution

Let $S$ be a compact algebraic surface with du Val singularities and $T$ be the minimal resolution of it. Is it true that
$$
H_2(T,\mathbb{Z})/(\oplus_{i}\mathbb{Z}E_i)\cong H_2(S,\mathbb{Z}),
$$
...

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### Cohomology and conifold transition for the quintic

Let $Y\subset \mathbb{C}P^4$ be the quintic threefold given by the equation $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4+5X_0X_1X_2X_3X_4=0$$
it has 125 singular points whose links are homeomorphic to $S^2\times ...

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### Do level sets always correspond to even graphs?

Suppose I have a level set of some function $f\colon\mathbb{R}^n\rightarrow\mathbb{R}^m$, say $L:=\{x:f(x)=c\}$. Let $S$ denote the points in $L$ at which $L$ is locally diffeomorphic to an open ...

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### Do there exist double points on an algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?

The title explains it all.
I'm familiar with the du val singularities on surfaces, also known as rational double points. In http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf, 2.1, they are ...

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### singularities of the dual variety of a surface

I am looking for a proof/reference of the following simple fact, which I think it holds true.
Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the ...

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### Which isolated surface singularity comes from a -5 curve?

Define the surface $X$ to be the total space of $\mathcal{O}_{\mathbb{P}^1}(-5)$.
By contracting the exceptional curve in $X$, we get a surface with an isolated singularity. I am looking for the ...

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### Obstruction map for local singularities via tangent (Andre-Quillen) cohomology

Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent ...

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328 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 ...