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8
votes
1answer
361 views

Analytical formula for topological degree

At the first page of the following article http://arxiv.org/pdf/1004.1018v1.pdf [edit: the formula on the arXiv differs from the formula in the published paper, and the formula displayed below is the ...
4
votes
1answer
354 views

Valuation of an ideal in a two-dimensional regular local ring

Let $f,g$ be two coprime elements in the ring $K[[x,y]]$, with $K$ a field. What is the smallest integer $n$ such that the inclusion of ideals $$(x^n)\subset (f,g)$$ holds in $K[[x,y]]$? Can we ...
5
votes
1answer
184 views

Infinitesimal deformations of a singular projective surface

Let $X$ be a normal projective surface with just two singular points $x_1,x_2\in X$, where $X$ has rational quotient singularities. Assume that both the singularities in $x_1$ and in $x_2$ admit a ...
5
votes
0answers
100 views

Framed singular knots

I've recently run across what one might (and I suspect people probably do) call framed singular knots, or maybe singular ribbon knots. Regardless of the name, what I mean is the following: Let $D$ be ...
0
votes
0answers
40 views

Singularities of the product of a $(\mathbb{C}^*)$-surface with $\mathbb{C}$

Recall that any normal $\mathbb{C}^*$-surface is Cohen-Macaulay and there exists normal $\mathbb{C}^*$-surfaces whose singularities are not rational. Does anyone know an example of a normal ...
1
vote
0answers
74 views

Is there any explicit result on the triangulated category of singularities of a curve?

This question is related to this MO question. Let $X$ be a projective curve over a field $\mathbb{C}$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category ...
7
votes
0answers
146 views

Singularities of an analytic function over a non-archimedean field

What do we know about the types of singularities that a convergent power series over a non-archimedean field can have? More specifically: i) What types of essential singularities can occur? ii) Are ...
1
vote
2answers
193 views

Cohen-Macaulayness of the direct image of the canonical sheaf

Let $Y$ be a normal projective variety and let $f:X\to Y$ be a desingularization. Define $\mathcal K_X=f_*\omega_X$, the Grauert--Riemenschneider canonical sheaf of $X$. It is independent of the ...
4
votes
1answer
657 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 ...
6
votes
4answers
1k views

minimal resolution of singularities

What is the minimal resolution of singularities of the surface $S^2(X^3+Y^3+Z^3)-3(S^2+T^2)XYZ=0$ which is a subset of $\mathbb{P}^1\times\mathbb{P}^2$ Please note that in this equation ...
4
votes
0answers
101 views

$\mathbb{Q}$-factoriality of singularities

I would like to understand if a certain variety is $\mathbb{Q}$-factorial (i.e., if every Weil divisor $D$ has a multiple $mD$ that is Cartier). This property can be deduced by a local picture around ...
3
votes
1answer
181 views

Analytically but not algebraically smoothable singularity

Are there examples of algebraic singularities which may be smoothed analytically but not algebraically? It certainly seems possible, but if not, why? Are there conditions under which this becomes ...
0
votes
2answers
122 views

Hochster-Roberts Theorem reciprocal

Given a Cohen-Macaulay ring $R$ over a field of characteristic zero and $G$ a reductive algebraic group acting on $R$, then the ring of ivanriants $R^G$ is also Cohen-Macaulay. This is known as ...
8
votes
2answers
289 views

Implicit Function Theorem on Singular Varieties

Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function $$f\colon X \to Y $$ and assume that it is bijective at the level of ...
0
votes
0answers
43 views

normality of truncated arc space

Let $X=Spec(A)$, with $A$ a normal $k$-algebra of finite type, $k$ is a field. For any integer $n$, let $X(k[t]/(t^{n}))$ the $n$-th truncated arc space, is it also normal? Same question for ...
8
votes
2answers
997 views

“Arithmetic genus” of a plane curve singularity.

I believe that the following questions are very basic, but I don't know how to get a reference. Consider a curve in the plane $C\in \mathbb C^2$ with a singularity at $0$ and suppose it is ...
2
votes
1answer
180 views

Is there a formula for the number of rational cuspidal curves in surfaces other than P^2?

Let $M$ be a two dimensional compact complex manifold and $A \in H_2(M, \mathbb{Z})$ a fixed homology class. Define a rational curve in $M$ to be $\textit{1-cuspidal}$ if the singularities of the ...
3
votes
0answers
122 views

Implicit function theorem for singularities

I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild". For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a ...
3
votes
0answers
205 views

Hypersurface with singularities

I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?). It was said ...
0
votes
0answers
80 views

How to understand/analyze vanishing cycles and fibers of 6 dimensional Lefschetz fibration?

Say you have a polynomial $f(x,y,z,m)=x^3+y^3+z^3+m^3$ where $x,y,z,m \in \mathbb{C}$. Consider the Lefschetz fibration from $\{f=\mu\} \cap \{ |m|\leq \delta \}$ to $m$ for suitably small $\mu$ and ...
1
vote
0answers
71 views

Global topological equivalence of Morse functions

Two Morse functions $f$ and $g$ are called topologicaly equivalent if there are diffeomorphism $h$ of the source and orientational-preserving diffeomorphism $k$ of the target such that $f=k\circ ...
1
vote
1answer
107 views

About 3-fold log canonical singularity

As far as I know, log canonical surface singularities were classified. How about higher dimensional case? I especially want to know whether given 3-fold singularity is log canonical or not. Let $f$ ...
0
votes
0answers
167 views

projective map from $\overline{\mathcal{M}}_{0,n}$

Suppose I have a morphism $f:\overline{\mathcal{M}}_{0,n} \to \mathbb{P}^N$ birational onto its image, and I know exactly what $F$-curves are contracted (or "dually", what divisors are contracted). ...
6
votes
0answers
84 views

When is a smooth function locally equivalent to a truncation of its Taylor series?

Let $U \subset \mathbb{R}^n$ be an open set and let $f:U \rightarrow \mathbb{R}$ be a smooth proper function. For $p \in \mathbb{R}^n$ let $$T_p(x_1,\ldots,x_n) = \sum_{i_1,\ldots,i_n=0}^{\infty} ...
3
votes
3answers
739 views

Poll about your proof of resolution of singularities and a request for advice

The questions first: What is the proof of resolution of singularities that you know? Why am I asking?: There are a number of proofs of resolution of singularities of varieties over a field of ...
0
votes
0answers
44 views

Disturbing regular level submanifold of a smooth function

Let $a$ be a regular value of a smooth function on a closed manifold and $\{f=a\}$ a corresponding level submanifold. It is known that any such function can be approximated by a Morse function $g$. ...
1
vote
0answers
78 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 ...
6
votes
1answer
532 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 $p_1,...,p_n$. ...
1
vote
1answer
196 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
1answer
173 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
0answers
177 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 ...
0
votes
0answers
109 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 ...
4
votes
0answers
108 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
1answer
255 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
0answers
59 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
2answers
268 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 ...
3
votes
1answer
90 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
1answer
225 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 ...
0
votes
0answers
74 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 ...
13
votes
1answer
403 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
0answers
124 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
308 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
311 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
440 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 ...
8
votes
1answer
308 views

Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...
1
vote
0answers
54 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
1answer
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 ...
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
263 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 ...
1
vote
0answers
64 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 ...