Questions tagged [resolution-of-singularities]
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190
questions
8
votes
1answer
152 views
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 ...
1
vote
0answers
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\...
4
votes
1answer
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 ...
3
votes
1answer
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
$$ (x,y,z) \to (\epsilon x, \epsilon y, \epsilon^2 z).$$...
3
votes
0answers
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 ...
1
vote
0answers
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 ...
5
votes
0answers
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 ...
0
votes
0answers
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 ...
1
vote
0answers
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 $...
0
votes
0answers
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 ...
6
votes
2answers
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
votes
1answer
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 ...
1
vote
0answers
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 ...
1
vote
1answer
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
2answers
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/...
1
vote
0answers
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 $...
2
votes
0answers
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 ...
6
votes
1answer
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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{...
9
votes
0answers
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 ...
2
votes
0answers
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 ...
1
vote
0answers
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 \...
3
votes
0answers
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 $S$, ...
6
votes
0answers
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'\...
2
votes
0answers
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 ...
2
votes
0answers
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 ...
1
vote
0answers
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 ...
7
votes
1answer
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 : (\...
7
votes
0answers
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
votes
0answers
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 ...
1
vote
1answer
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}^...
1
vote
0answers
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
vote
1answer
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 ...
3
votes
0answers
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?
2
votes
0answers
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 $R^G$ ...
1
vote
0answers
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}(...
1
vote
1answer
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,...
3
votes
0answers
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
1answer
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 ...
3
votes
0answers
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 ...
12
votes
0answers
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, ...
0
votes
0answers
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\...
1
vote
1answer
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 $\...
1
vote
0answers
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
votes
1answer
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 ...
4
votes
1answer
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 ...
3
votes
0answers
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 $C$?
1
vote
1answer
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
votes
1answer
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 ...
