All Questions
Tagged with resolution-of-singularities ag.algebraic-geometry
224 questions
1
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139
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Simultaneous Log resolutions for both varieties and divisors
Let $X$ be a normal variety and $D \subset X$ be a prime divisor which is also normal. It is well-known that we can take a resolution $f: W \to X$ of $X$ such that
$$\DeclareMathOperator{\Supp}{\...
- 3,472
8
votes
1
answer
273
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Is there a "minimal" Whitney stratification of a complex hypersurface?
Let $X\subset \mathbb C^n$ be a complex hypersurface (given by $F=0$ where $F$ is a polynomial). It is known then that $X$ admits a Whitney stratification. This is a decomposition of $X$ into smooth ...
- 14.3k
1
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242
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Smooth normalization and blow-up of the exceptional locus
Let $n:\widetilde X\rightarrow X$ be the normalization of a complex (quasi-projective) variety $X$. Assume $\widetilde X$ is smooth, that $n$ is an isomorphism outside a smooth connected subvariety $Y\...
- 1,463
5
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1
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355
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Computing the invariants of ball quotient surfaces
The two-dimensional complex unit ball $B$ has group of biholomorphic automorphisms $PU(2,1)$.
If $Γ$ is an arithmetic subgroup of $PU(2,1)$, the quotient $Γ\text{\\}B$ is an orbifold.
Taking its ...
- 9,501
4
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1
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509
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Resolution of 3-fold quotient singularities
This is exercise 1.10 from Reid's Young person's guide to canonical singularites.
Let $X=\mathbb{C}^3/ \mu_3$ where $\epsilon \in \mu_3$ acts by
$$ (x,y,z) \to (\epsilon x, \epsilon y, \epsilon^2 z).$$...
- 231
3
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0
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216
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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 ...
- 31
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93
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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 ...
- 91
6
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234
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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 ...
- 1,981
0
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0
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218
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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 ...
- 1,463
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90
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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 $...
- 1,463
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195
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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 ...
- 1,463
19
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1
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2k
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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 ...
- 199
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192
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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 ...
- 5,901
7
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2
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1k
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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/...
- 4,428
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200
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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 $...
- 231
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220
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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 ...
user114666
3
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0
answers
992
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 ...
- 623
9
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0
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361
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 ...
user158636
2
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0
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170
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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 ...
- 282
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410
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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 \...
- 6,006
4
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0
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125
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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$, ...
- 61
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388
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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'\...
- 1,803
2
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0
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85
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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 ...
- 393
2
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0
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224
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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 ...
- 393
1
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0
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178
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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 ...
- 393
7
votes
1
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231
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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),$$ where $f_i : (\...
user150450
7
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0
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680
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Artin's "On isolated rational singularities of surfaces"
My question deals with Michael Artin's paper "On isolated rational singularities of surfaces"; more precisely the proof of Theorem 4 on page 133. Here the relevant excerpt:
The Setting: Let ...
- 6,006
4
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0
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119
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Are there any explicit (prime-to-l) alterations for interesting varieties (or schemes)?
I have read that it is easier to find regular alterations of varieties than their resolutions of singularities (moreover, I believed in this sentence when I read it). My question is: do there exist ...
- 16.9k
2
votes
1
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317
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Singular locus of a linear system of hyperplane sections
Let $X\subset\mathbb{P}^N$ be a rational smooth projective irreducible non degenerated variety of dimension $n=\dim(X)$ and let $$\mathcal{H}=|\mathcal{O}_X(1) \otimes\mathcal{I}_{{p_1}^2,\dots,{p_l}^...
- 1,343
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927
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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 ...
- 623
1
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1
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216
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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 ...
- 2,126
2
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0
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163
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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?
user61586
2
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61
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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 $R^G$ ...
- 407
1
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0
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125
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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}(...
- 4,189
1
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1
answer
160
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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,...
- 6,006
4
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0
answers
320
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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 ...
- 489
3
votes
1
answer
190
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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
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0
answers
784
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 ...
- 6,006
14
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884
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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, ...
- 141
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0
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82
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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
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1
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247
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 $\...
- 2,126
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0
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93
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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 ...
- 375
3
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1
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470
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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 ...
user125056
2
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0
answers
164
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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 $C$?
user56259
1
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1
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383
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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 ...
- 623
6
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1
answer
1k
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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 ...
- 471
4
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0
answers
116
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Bertini-type theorem for strict transform
Let $(X,o)$ be an isolated, normal singularity of dimension at least $3$. Let $\pi: \widetilde{X} \to X$ be a resolution of singularity of $X$. Is it true that for a general hypersurface $H \subset X$ ...
- 2,323
3
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0
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452
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Singularities of rational quartic surfaces
Let $X\subset \mathbb{P}^3$ be an irreducible quartic surface, defined over an algebraically closed field $k$. Suppose that $X$ is rational (i.e. birational to $\mathbb{P}^2$). Is is true that $X$ has ...
- 7,722
3
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0
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167
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Examples of explicit computations of log-resolutions
I have been working with log-resolutions lately and learning more about them. I am aware that in general producing explicit log-resolutions is difficult, but I was wondering if this has been done in ...
8
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1
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496
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Is canonical model always with canonical singularity
Let X be a smooth variety, take the proj of canonical ring of X and denote it by Y. Is Y always a canonical variety? I know it's true for general type variety. Thank in advance.
- 623