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0
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1answer
132 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
1answer
68 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
0answers
62 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 ...
1
vote
0answers
118 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
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1answer
305 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
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0answers
109 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
3answers
215 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
0answers
264 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
2answers
271 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
0answers
47 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
0answers
46 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
0answers
99 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 ...
1
vote
1answer
228 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
1answer
253 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
1answer
264 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
1answer
128 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
0answers
113 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
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0answers
68 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
2answers
232 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
1answer
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
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0answers
74 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
1answer
136 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
1answer
200 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
1answer
129 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
1answer
394 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
1answer
171 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
0answers
304 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
0answers
64 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?
5
votes
1answer
386 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
1answer
96 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
0answers
223 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
1answer
204 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
0answers
29 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
1answer
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
0answers
117 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
2answers
225 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
1answer
259 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
1answer
162 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
1answer
207 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
0answers
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
1answer
363 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
2answers
248 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 ...
3
votes
3answers
178 views

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. ...
0
votes
0answers
117 views

Action on C[[X,Y]]/f(X,Y) giving complete intersection quotients

Let $R \colon\!= {\Bbb C}[[X,Y]]/(f(X,Y))$ be a complete local ring of Krull-dimension $1$. Assume that we have an action of $\Bbb Z$ on $R$ such that fixed elements by $\Bbb Z$ in $R$ are only ...
3
votes
0answers
183 views

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 ...
0
votes
0answers
55 views

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 $$ H_2(T,\mathbb{Z})/(\oplus_{i}\mathbb{Z}E_i)\cong H_2(S,\mathbb{Z}), $$ ...
8
votes
0answers
260 views

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 ...
9
votes
1answer
238 views

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 ...
3
votes
1answer
244 views

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 ...
0
votes
1answer
218 views

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 ...