# Questions tagged [resolution-of-singularities]

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190
questions

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### Is there a “minimal” Whitney stratification of a complex hypersurface?

Let $X\subset \mathbb C^n$ be a complex hypersurface (given by $F=0$ where $F$ is a polynomial). It is known then that $X$ admits a Whitney stratification. This is a decomposition of $X$ into smooth ...

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100 views

### Smooth normalization and blow-up of the exceptional locus

Let $n:\widetilde X\rightarrow X$ be the normalization of a complex (quasi-projective) variety $X$. Assume $\widetilde X$ is smooth, that $n$ is an isomorphism outside a smooth connected subvariety $Y\...

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157 views

### Computing the invariants of ball quotient surfaces

The two-dimensional complex unit ball $B$ has group of biholomorphic automorphisms $PU(2,1)$.
If $Γ$ is an arithmetic subgroup of $PU(2,1)$, the quotient $Γ\text{\\}B$ is an orbifold.
Taking its ...

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**1**answer

142 views

### Resolution of 3-fold quotient singularities

This is exercise 1.10 from Reid's Young person's guide to canonical singularites.
Let $X=\mathbb{C}^3/ \mu_3$ where $\epsilon \in \mu_3$ acts by
$$ (x,y,z) \to (\epsilon x, \epsilon y, \epsilon^2 z).$$...

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102 views

### When do crepant resolutions of quotients of Calabi-Yau varieties exist?

Suppose I have a Gorenstein variety $X$ over $\mathbb{C}$ with trivial canonical bundle, and the action of a finite group $G$ on $X$, which acts trivially on the canonical bundle.
Question. When does ...

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46 views

### Partial crepant resolution in codimension 2

Let $\xi_5$ be a 5-root of the unity. We consider $\mathbb{C}^4/G$, where $G=\left\langle \sigma,\tau\right\rangle$, with $\sigma$ and $\tau$ the automorphisms given, respectively, by the following ...

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134 views

### Resolution graph of higher dimensional ADE singularities

I am looking for different configurations of the exceptional divisors arising from blowing up a higher dimensional ADE singularity (see p. 240 of this article of Bruns for a description of such ...

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115 views

### Normality of a blow up

Let $X$ be a quasi-projective normal $\mathbb C$-variety and $Y\subset X$ a smooth subvariety such that $X$ is normally flat along $Y$ (i.e. the normal cone of $X$ along $Y$ is flat over $Y$).
Is the ...

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43 views

### Blowing-up the exceptional locus of a (small) resolution

Let $f:\widetilde X\rightarrow Y$ be a proper morphism between smooth complex varieties which is birational unto its image $X=f(X)$. Assume the singular locus $W\subset X$ of $X$ is smooth and that $...

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122 views

### Nice small resolution and normality of blow-up

Let $X$ be a complex variety whose singular locus is a smooth variety $Z$.
Let $f:Y\rightarrow X$ be a small resolution of $X$ such that $f^{-1}(z)$ is smooth for any $z\in Z$ and $\dim(f^{-1}(z))$ is ...

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266 views

### Symplectic resolutions amongst cotangent bundles

It is known that a generalized flag variety $X=T^*(G/P)$ is a (symplectic) resolution of singularities of its affinization $X^\text{aff}\mathrel{:=}\operatorname{Spec}(H^0(X,\mathcal{O}_X))$. In type ...

**18**

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1k views

### Perfectoid approach to resolution of singularities in char $p$

Since perfectoid techniques have built a bridge between char $0$ and char $p$ worlds, it is conceivable that they can be applied to resolution of singularities in char $p$ using their successful ...

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106 views

### Factorization of birational maps in char $p$

So I was reading about the factorization result that any birational map between smooth varieties is composition of blow-ups and blow-downs with smooth centers. It is apparently true only in ...

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**1**answer

89 views

### Realizing a set as the image of a smooth map

Consider the following subset of $\mathbb{R}^2$:
$S = \{ (x, y) \in \mathbb{R}^2 : |y| \leq |x|^{3/2} \}$
(See here for a plot on Wolfram Alpha.)
The origin $(0, 0)$ is a kind of singular point of $S$....

**5**

votes

**2**answers

528 views

### Cohomology of resolution of singularity

If $X,Y$ are smooth projective schemes, then if we have a surjection $f:X\to Y$, we have an injective map on étale cohomology, or more generally on any Weil cohomology (see https://mathoverflow.net/q/...

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91 views

### Second chern class of a rank 2 bundle

Here is another question from the paper "Second Chern class and Riemann-Roch for vector bundles on resolutions of surfaces singularities" by J. Wahl. This is at the beginning of the section $...

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89 views

### Divisorial contraction to a non-normal variety

Consider a divisorial contraction $f:X\rightarrow Y$, between projective varieties, contracting an irreducible divisor $D\subset X$ to a subvariety $Z\subset Y$ of codimension at least two, and which ...

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168 views

### Singularities of PL embedding of surface in a contractible 4-manifold

I am trying to understand the article "A solution to a conjecture of Zeeman" by Akbulut, but I am not an expert in PL-geometry.
As far as I understand, two statements should be true, but I ...

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200 views

### Definition of Q gorenstein variety

I have a question about the definition of Q-Gorenstein variety.
I saw a definition of Q-Gorenstein variety：for a normal variety $X$, it's Q-Gorenstein if the canonical divisor is Q Cartier. I wonder ...

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51 views

### Strict transform does not modify the Normalization

I have a question about a reduction step in proof of Lemma 10.1.24 from Qing Liu's Algebraic Geometry and Arithmetic Curves on page 463:
Firstly we reduce to $S$ local, but then we replace $\mathcal{...

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209 views

### Would full resolution of singularities have cohomological implications beyond the alteration theory?

De Jong's result on alterations allows one to show the potential semistability of certain Galois representations arising from cohomology of varieties (among other things). If we knew the existence of ...

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136 views

### Resolution of pairs in characteristic p

Let $R$ be a complete DVR of characteristic $p$, say $R=\mathbb{F}_p[[t]]$, and $X$ be a reduced scheme of finite type over $R$. Let also $X_s$ denote the special fiber of $X$. If I understand ...

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185 views

### Separable morphism of curves

A proof from Janos Kollar's Lectures on Resolution of Singularities Kollar (p 37) works as follows:
Theorem 1.58 (M. Noether, 1871). Let $k$ be an algebraically closed
field and $C \subset \...

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57 views

### Embedded normalization

Let $S$ be an irreducible surface in a 3-dimensional variety $X$ (everything taking place over $\mathbb{C}$, say). By Hironaka's therorem, we know for sure that there is an embedded resolution of $S$, ...

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243 views

### Globalization of Brieskorn-Grothendieck resolution

Brieskorn (1970) showed that for semiuniversal deformation of rational double points surface singularities $X\to S$, there is a finite base change $S'\to S$, such that the new family $f:X\times_{S}S'\...

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73 views

### Is toroidalization local?

Let $f:X \to Y$ be a surjective morphism of smooth projective varieties, $D$ be a simple normal crossings divisor on $X$ and $U_Y \subset Y$ be an open subset over which $(X,D)$ is log smooth (in the ...

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82 views

### Log canonical centers of toric (and toroidal) varieties

Q1: Let $(X,B)$ be a toric variety. There exists a toric resolution of singularities $f:(Y,E) \to (X,B)$. Here is my question:
Is any lc center of $(X,B)$ an irreducible component of an intersection ...

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95 views

### Birational model of a log smooth pair

Given a log smooth pair $(X,B)$ with a reduced boundary divisor $B$, consider a birational model $\pi:X' \to X$ and a boundary divisor $B'$ which is given by $K_{X'}+B'=\pi^*(K_X+B)$. Here is my ...

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206 views

### Resolution graphs in the sense of Némethi

The following definitions are from lecture notes of Némethi. A surface singularity $(X,0)$ is defined by $$(X,0) = (\{ f_1 = \ldots = f_m=0 \}) \subset \mathbb (\mathbb{C}^n,0),$$ where $f_i : (\...

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451 views

### Artin's “On isolated rational singularities of surfaces”

My question deals with Michael Artin's paper "On isolated rational singularities of surfaces"; more precisely the proof of Theorem 4 on page 133. Here the relevant excerpt:
The Setting: Let ...

**4**

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107 views

### Are there any explicit (prime-to-l) alterations for interesting varieties (or schemes)?

I have read that it is easier to find regular alterations of varieties than their resolutions of singularities (moreover, I believed in this sentence when I read it). My question is: do there exist ...

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**1**answer

155 views

### Singular locus of a linear system of hyperplane sections

Let $X\subset\mathbb{P}^N$ be a rational smooth projective irreducible non degenerated variety of dimension $n=\dim(X)$ and let $$\mathcal{H}=|\mathcal{O}_X(1) \otimes\mathcal{I}_{{p_1}^2,\dots,{p_l}^...

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276 views

### canonical divisor on singular curves with nodal point

What's the definition of canonical divisor(or whatever related concept) on singular curve with nodal point. More generally, what the definition of canonical divisor on a singular variety X, which is ...

**1**

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**1**answer

162 views

### Pushforward of locally free sheaves and resolution of singularities

Let $X$ be an noetherian, affine, isolated of dimension at least $2$ and $\pi:\widetilde{X} \to X$ a resolution of singularities. Let $\mathcal{E}$ be a locally-free sheaf on $\widetilde{X}$ such that ...

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108 views

### Terminal and log canonical singularities

Let $D$ be a divisor with at most terminal singularities in a smooth projective variety $X$. Is the pair $(X,D)$ log canonical?

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54 views

### Test rational singularities after forming invariants

Let $R$ be a normal local domain of dimension $2$ and odd residue characteristic endowed with the action of the finite group $G \cong \mathbb{Z}/2\mathbb{Z}$. Suppose that the ring of invariants $R^G$ ...

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110 views

### Resolution of rational surfaces

Let $S$ be a rational singular complete algebraic surface over $\mathbb{C}$. Let $\phi:\tilde{S}\to S$ be a resolution of singularities with minimal possible Picard rank (i.e. minimal $\mathrm{dim}(...

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**1**answer

117 views

### Generators of a graded algebra defining bundle over elliptic curve

I have a question about a statement from P. Wagreich's paper "Elliptic Singularities of Surfaces" (page 425):
We consider an elliptic curve $X$ and a line bundle (=invertible sheaf) $L$ on $X$.
Then,...

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149 views

### Toric Fan for the Du Val's singularities D_n and E_n

Let us consider the Du Val's singularities.
i.e. https://en.wikipedia.org/wiki/Du_Val_singularity.
It is well known that they are classified by ADE, because the exceptional divisors arising in the ...

**3**

votes

**1**answer

155 views

### Igusa zeta functions of univariate polynomials: $\mathbb{Z}_p$ or $\mathbb{Q}_p$ in this statement

Let $f\in\mathbb{Z}_p[X]$ and let $Z_{f,p}(T)\in\mathbb{Z}_{(p)}(T)$ be the $p$-adic Igusa zeta polynomial (i.e. $Z_{f,p}(p^{-s})$ is the $p$-adic Igusa zeta function in the complex variable $s$, with ...

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291 views

### Intuition behind RDP (Rational Double Points)

Let $S$ be a surface (so a $2$-dimensional proper $k$-scheme) and $s$ a singular point which is a rational double point.
One common characterisation of a RDP is that under sufficient conditions there ...

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749 views

### On mixed $p$-adic Hodge theory

Does mixed $p$-adic Hodge theory exist? Can we extend the scope of comparison theorems using simplicial resolutions a la Deligne? Do we get 3 opposite filtrations as in classical mixed Hodge theory, ...

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58 views

### Bijective restriction of the normalization morphism

Let $X$ be an integral separated scheme of finite type over $\mathbb{C}$. Consider the normalization morphism $f:X'\rightarrow X$. Can we always find an affine open $U\subset X'$ such that $f|_U:U\...

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**1**answer

174 views

### Infinitesimal deformation of strict transform

Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\...

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67 views

### Blow up of 9 points in 3-fold and intersection of strict transforms

Suppose we have blown up a variety $X$ at some points $P_j$ so that we introduce exceptional divisors $E_j$ in $\widetilde X$; what is the general strategy to determine the intersections of these ...

**4**

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101 views

### Image of a quiver variety under natural morphism

We know that the natural morphism $\pi:\mathfrak{M}_{\theta}(Q,\mathbf{v},\mathbf{w})\rightarrow \mathfrak{M}_0(Q,\mathbf{v},\mathbf{w})$ between a smooth and affine quiver variety is not necessarily ...

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**1**answer

230 views

### Automorphisms of singular hypersurfaces

Let $X\subset\mathbb{P}^{n+1}$ be an irreducible and reduced hypersurface of degree $d$.
A theorem by Matsumura and Monski asserts that if $n\geq 2$, $d\geq 3$, $(n,d)\neq (2,4)$ and $X$ is smooth ...

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116 views

### Cubic 3-fold singular along a curve

Does there exists a cubic or quartic $3$-fold $X\subset\mathbb{P}^4$ such that $Sing(X)$ is a smooth curve $C$ of genus $g(C)\geq 2$ and $X$ has $A_1$-singularities along $C$?

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208 views

### Construction of log canonical singularity

I know there's classification about normal log canonical surface singularity in the sense of configuration of exceptional curves.
There is one type of log canonical singularity(not klt) whose ...

**3**

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377 views

### Some naive questions on crepant resolutions of singularities

I have some clarifying questions about crepant resolutions of singularities. The definition that I am aware of is that if $f:\tilde X \to X$ is a resolution of singularities, then the resolution is ...