The singularity-theory tag has no usage guidance.

**7**

votes

**1**answer

238 views

### Cohomology of tangent sheaf of a singular hypersurface

Let $X\subset\mathbb{P}^n$ be a hypersurface singular at finitely many points $p_i\in X$. We may assume that $X$ has ordinary singularities at the $p_i$'s.
Does there exists a formula, perhaps in ...

**2**

votes

**1**answer

155 views

### Kahler Ricci flow in Fano fibration

Let $f:X\to Y$ be a Fano fibration of Kahler manifolds $X, Y$. Then why the Kahler Ricci flow
$$\frac{\partial \omega}{\partial t}=-Ric(\omega(t))$$
starting of $[\omega_0]=f^*(\omega_Y)+c_1(X)$ ...

**4**

votes

**1**answer

164 views

### (Etale) fundamental group of quotient singularity $\mathbb{C}^n/G$

I don't know much about (algebraic/etale) fundamental groups, so sorry if this question sounds stupid. I am interested in quotient singularities (quotients $X$ of $\mathbb{C}^n$ by a finite subgroup ...

**6**

votes

**1**answer

575 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 ...

**0**

votes

**0**answers

38 views

### When is a critical value of a map contained in the interior of the image?

Let $M^n$ be a compact manifold, and $F\colon M \to \mathbb{R}^n$ a smooth map. The inverse function theorem implies that every regular value of $F$ lies in the interior of $F(M)$, hence every point ...

**1**

vote

**0**answers

86 views

### Does the link of a hypersurface singularity determine its analytic type?

Consider a hypersurface $V(f) \subseteq \mathbb{C}^{n+1}$ with an isolated singularity at the origin. If $L := V(f) \cap S^{2n+1}_\epsilon$ is the link of $V(f)$ (with $S^{2n+1}_\epsilon$ a ...

**4**

votes

**0**answers

167 views

### Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical

Let start with a definition
Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So ...

**1**

vote

**1**answer

163 views

### General Reference for surface singularities

Is there any "standard" reference for (rational) singularities on algebraic surfaces? I'm aware of Artin's papers and the one of Brieskorn (Rationale Singularitäten komplexer Flächen), but they seem ...

**4**

votes

**1**answer

96 views

### The volume around a singular isolated root when equalities are loosened

Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a ...

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votes

**0**answers

165 views

### Canonical scheme structure on the singular locus of a variety

I first asked this question on Math StackExchange but no answers were given.
Let $X$ be subvariety of affine space $\mathbb{A}_{k}^n$, where $k$ is a field, and suppose $X$ is given by equations
...

**2**

votes

**0**answers

76 views

### Singularities of algebraic curves, and torsion of the pull-back of the differential module by the normalisation

The problem in the following :
given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field ...

**2**

votes

**0**answers

157 views

### Ricci curvature in resolution of singularities

Let $X$ and $X'$ are Kahler variety and $f: (X',\omega')\to (X,\omega)$ be the resolution of singularities of $X$ then from $K_X=f^*K_X'+E$ how can we find
the relation between $Ric(\omega)$ and ...

**0**

votes

**0**answers

132 views

### A definition of arithmetic divisor with conic singularities?

I have a question related to the preprint "Heights and metrics with logarithmic singularities" by G. Freixas i Montplet.
Let $X$ be an arithmetic variety with arithmetic divisor $D$ how can we ...

**4**

votes

**1**answer

233 views

### Complexifying a real-analytic singularity

This is probably a well-known issue, but I could not find a clear discussion in the literature, and I think others could find it useful.
Consider a real-analytic function germ $f:(\mathbb R^2,0) ...

**2**

votes

**1**answer

243 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) ...

**6**

votes

**1**answer

223 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 ...

**2**

votes

**1**answer

348 views

### Reference request: English translation of Brieskorn 1970 paper

Is there any english (or french) translation of the following paper by Brieskorn (1970)?
Brieskorn, E., "Die Monodromie der Isolierten Singularitäten von Hyperflächen", Manuscripta Mathematica 2 ...

**3**

votes

**1**answer

179 views

### Rationality of higher dimensional du Val singularities

I am interested in the isolated singularity defined over $\mathbb{C}$ by
$$
x_1^2+\cdots + x_n^2+x_{n+1}^k=0,
$$
where $n>2$ and $k>2$.
I would like to know whether this singularity is ...

**3**

votes

**0**answers

83 views

### Is there a correspondence between counting curves in P^2 blown up at a point and counting curves in P^2?

Let $X$ be $\mathbb{P}^2$ blownup at one point
and $\beta := d L -2E \in H_2(X, \mathbb{Z})$, where $L$ and $E$
denote the class of a line and the exceptional divisor respectively.
Let ...

**10**

votes

**2**answers

526 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 ...

**5**

votes

**0**answers

112 views

### Gysin exact sequence for a singular subvariety

Let $k$ be an algebraically closed field (I'm interested in a characteristic $p>0$ specific example) and let $X$ be a (smooth if needed) algebraic variety.
Let $Y \subset X$ be a (possibly) ...

**1**

vote

**0**answers

146 views

### reference for weighted blow-up

Let $(0\in X)$ be a germ of a normal 3-fold with a singular point $0$
(over $\mathbb{C}$).
We think of $X$ as a small neighborhood of $0$ (for studying singularity).
If we can think $X$ as a ...

**3**

votes

**1**answer

302 views

### Stratification of complex algebraic varieties

Let $V$ be a complex quasi-projective variety, we know from H. Whitney's and B Teissier works on stratifications of algebraic varieties that $V$ has an intrinsic stratification
$$X_0\subset ...

**2**

votes

**0**answers

86 views

### Is it obvious that the defining conditions to obtain a particular singularity are well-defined on the quotient space?

Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function
vanishing at the origin, with
the following properties:
$$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...

**1**

vote

**0**answers

70 views

### Is a variety a local complete intersection if it is locally a complement of to a smooth $N$-dimensional affine of $N-m$ affine subvarieties?

If an equidimensional variety $V$ of dimension $m$ is locally a set-theoretic complete intersection (i.e., it can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in ...

**2**

votes

**0**answers

66 views

### quasi-ordinary singularities on a versal deformation?

Let $V$ be a variety over $\mathbb{C}$ and suppose $O$ is a singular point of $V$. Are there conditions on $(V,O)$ such that a versal deformation $W$ of $(V,O)$ has only quasi-ordinary singularities.
...

**9**

votes

**2**answers

452 views

### Which weighted projective spaces (and their finite quotients) are local complete intersections?

Let $G$ be a finite subgroup of $\textrm{Gl}_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of ...

**2**

votes

**0**answers

395 views

### Weil Petersson metric on moduli space of Calabi Yau manifolds

Let $f:(X,D)\to Y$ be a holomorphic fibre space where $D$ is divisor with conic singularities and let fibres $(X_s,D_s)$ are log Calabi-Yau pair .i.e $K_X+D$ is nummerically trivial, then we have ...

**3**

votes

**1**answer

174 views

### Jaffe's exact sequence

Let $X$ be a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. Consider the exact sequence $$0 \rightarrow \text{Pic}(X) \rightarrow ...

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votes

**0**answers

92 views

### Toric Degenerations and Nearby Cycles

Suppose that $f: X \to \mathbb{A}^1$ is a toric degeneration in the sense of Nishinou-Siebert. In other words let X be a (possibly singular) toric variety equipped with a (not necessarily proper) ...

**1**

vote

**0**answers

77 views

### If there exists an immersion, then does a neighbourhood of a singular rational curve contain a genuine cuspidal point?

Let $X$ be a compact complex surface and $u_1, u_2: \mathbb{P}^1 \longrightarrow X$ be rational curves that are not multiply covered that represents a class $\beta \in H_2(X, \mathbb{Z})$. Suppose ...

**0**

votes

**0**answers

66 views

### Characterization of Singular locus

Let A be a complete regular local ring over a field k and B be a complete normal local ring over a field k. We assume that (Krull-dimension of A) > 1.
We consider the ring homomorphism f: A ---> B, ...

**6**

votes

**1**answer

217 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

**0**answers

113 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

**0**answers

43 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

**0**answers

90 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

**0**answers

170 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

**2**answers

204 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

**1**answer

758 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

**4**answers

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

**0**answers

122 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

**1**answer

201 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

**2**answers

140 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

**2**answers

342 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 ...

**8**

votes

**2**answers

1k 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

**1**answer

199 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

**0**answers

153 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

**0**answers

228 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 ...

**1**

vote

**0**answers

78 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

**1**answer

119 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 $f$ ...