# Tagged Questions

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82 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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64 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 ...

**3**

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**1**answer

253 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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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 ...

**1**

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**1**answer

144 views

### 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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116 views

### Hypersurfaces with Gorenstein singular loci

Recall that a hypersurface $D$ in a complex manifold $X$ is called a free divisor if the Lie algebroid $\mathcal{T}_X(-\log D)$ of vector fields tangent to $D$ is locally free. This condition is ...

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62 views

### semicontinuity of the conductor defined by Temkin

We say a principal pair $(X,\mathcal{I})$ where $X=Spec(A)$ is affine scheme and $\mathcal{I}=\tilde{I}$ where $I\subset A$ is a principal ideal generated by $\pi$ wich is a non zero divisor.
For a ...

**5**

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**1**answer

147 views

### Is there an algorithm to find out the number of small solutions to a polynomial equation, when we vary all the coefficients?

Let $\Phi (z,t)$ be a polynomial given by
$$ \Phi(z,t) := z^n + A_{n-1}(t) z^{n-1} + \ldots + A_1(t) z + A_0(t).$$
Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation
$$ ...

**7**

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**1**answer

358 views

### Factoriality: local or global?

Let $X$ be an algebraic variety. I have read the following definitions:
$X$ is factorial if every Weil divisor on $X$ is Cartier.
$X$ is locally factorial if all its local rings are unique ...

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330 views

### Jacobian ideals reference

Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ...

**6**

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**1**answer

361 views

### Factoriality vs $\mathbf{Q}$-factoriality for threefolds hypersurfaces with isolated singularities

Let $X \subset \mathbf{P}^4$ be a complex threefold hypersurface with isolated singularities.
We denote as usual by $\textrm{Cl}(X)$ the group of Weil divisors modulo linear equivalence and by ...

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442 views

### An elementary question in singularities

The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ...

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**2**answers

366 views

### on the relative conductor of curve singularity and quotient of ideals

Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ ...

**12**

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**4**answers

970 views

### What formal properties should resolution of singularities have?

If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...

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871 views

### Matrix factorization categories for ADE singularities

What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated.
Background: For ADE singularities, see for example this. For ...

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**2**answers

1k views

### Limit of a series of singularities

The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities ...