All Questions
Tagged with resolution-of-singularities ag.algebraic-geometry
224 questions
4
votes
1
answer
394
views
References on Namikawa-Weyl group
What are the most reasonable references on the definition of the Namikawa-Weyl groups and the first results about them?
In particular, are there more recent (or more educational) texts than the ...
- 4,612
2
votes
1
answer
238
views
Preimage by birational maps
I am looking for an example (I guess that in complex projective space $\mathbb{P}^{n}$ is good) such that satisfy the following condition (in non trivial case, for this assume $X \neq \tilde{X}$):
Let ...
- 7,320
2
votes
0
answers
220
views
plumbing description of resolution of ADE singularities
Let $G$ be a finite subgroup of $SU(2)$ and consider the quotient of the unit ball $B\subset \mathbb{C}^{2}$ by $G$. The result, denoted by $V$, has a boundary $S^{3}/G$ and has an ADE singularity at $...
- 1,069
7
votes
1
answer
206
views
$2$-dimensional complete local normal domain with rational singularity that has exactly one exceptional curve
Let $(R, \mathfrak m)$ be a complete local normal domain of dimension $2$ with residue field $R/\mathfrak m$ algebraically closed and characteristic $0$. Assume Spec$(R)$ has rational singularity, let ...
- 66.3k
3
votes
0
answers
170
views
Does existance of crepant resolution of tangent space imply existance of crepant resolution globally in the algebraic setting?
Suppose $X$ is smooth proper algebraic $\mathbb C$-variety with algebraic action of a finite abelian group $G$. Suppose I know that
$X/G$ (good geometric quotient) exists and it is normal Gorenstein ...
- 1,307
10
votes
1
answer
2k
views
Equivariant resolution of singularities
I am looking for some references on equivariant resolution of singularities. In most references quoted on mathoverflow (for instance : Reference on an equivariant resolution of singularities), they ...
- 63.9k
1
vote
0
answers
139
views
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
$$\DeclareMathOperator{\Supp}{\...
- 6,377
8
votes
1
answer
273
views
Is there a "minimal" Whitney stratification of a complex hypersurface?
Let $X\subset \mathbb C^n$ be a complex hypersurface (given by $F=0$ where $F$ is a polynomial). It is known then that $X$ admits a Whitney stratification. This is a decomposition of $X$ into smooth ...
- 7,320
4
votes
1
answer
509
views
Resolution of 3-fold quotient singularities
This is exercise 1.10 from Reid's Young person's guide to canonical singularites.
Let $X=\mathbb{C}^3/ \mu_3$ where $\epsilon \in \mu_3$ acts by
$$ (x,y,z) \to (\epsilon x, \epsilon y, \epsilon^2 z).$$...
- 701
1
vote
0
answers
242
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\...
- 1,463
5
votes
1
answer
355
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 ...
- 591
7
votes
1
answer
2k
views
Crepant resolutions of cDV singularities?
Compound Du Val 3-fold singularities form a good class of singularities in 3-fold singularity theory. I would like to know which singularities admit crepant resolutions. If I remember correctly, $cA_{...
- 2,310
3
votes
0
answers
216
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 ...
- 66.3k
19
votes
1
answer
2k
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 ...
- 21.3k
3
votes
1
answer
776
views
On the coherence of a Néron-ring
Let $A:= \underset{\lambda \in \Lambda}{\varinjlim} \,A_{\lambda}$ be an inductive limit of geometric regular local ring $(A_{\lambda}, {\frak m}_{\lambda})$, whose transition map $\phi_{\mu\lambda} \...
CommunityBot
- 1
1
vote
0
answers
93
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 ...
- 91
6
votes
0
answers
234
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 ...
- 1,981
0
votes
0
answers
218
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 ...
- 1,463
0
votes
0
answers
195
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 ...
1
vote
0
answers
90
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 $...
- 1,463
1
vote
0
answers
191
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 ...
- 5,901
7
votes
0
answers
680
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 ...
- 5,966
7
votes
2
answers
1k
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/...
- 180
1
vote
0
answers
199
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 $...
- 231
2
votes
0
answers
220
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 ...
- 63.9k
3
votes
4
answers
3k
views
Cone over the Veronese surface
Let $V\subset\mathbb{P}^5$ be the Veronese surface and let $X\subset\mathbb{P}^6$ be the cone over it. Since $X$ is $\mathbb{Q}$-factorial there are two integers $a,b$ such that $aK_X = \mathcal{O}_X(...
- 42.9k
7
votes
1
answer
770
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 ...
- 38k
3
votes
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
votes
0
answers
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 ...
CommunityBot
- 1
2
votes
0
answers
170
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 ...
- 12.9k
1
vote
0
answers
410
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 \...
- 5,966
7
votes
1
answer
231
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),$$ where $f_i : (\...
- 10.9k
4
votes
0
answers
125
views
Embedded normalization
Let $S$ be an irreducible surface in a 3-dimensional variety $X$ (everything taking place over $\mathbb{C}$, say). By Hironaka's therorem, we know for sure that there is an embedded resolution of $S$, ...
- 61
6
votes
0
answers
388
views
Globalization of Brieskorn-Grothendieck resolution
Brieskorn (1970) showed that for semiuniversal deformation of rational double points surface singularities $X\to S$, there is a finite base change $S'\to S$, such that the new family $f:X\times_{S}S'\...
- 1,803
2
votes
0
answers
85
views
Is toroidalization local?
Let $f:X \to Y$ be a surjective morphism of smooth projective varieties, $D$ be a simple normal crossings divisor on $X$ and $U_Y \subset Y$ be an open subset over which $(X,D)$ is log smooth (in the ...
- 393
2
votes
0
answers
224
views
Log canonical centers of toric (and toroidal) varieties
Q1: Let $(X,B)$ be a toric variety. There exists a toric resolution of singularities $f:(Y,E) \to (X,B)$. Here is my question:
Is any lc center of $(X,B)$ an irreducible component of an intersection ...
- 393
1
vote
0
answers
178
views
Birational model of a log smooth pair
Given a log smooth pair $(X,B)$ with a reduced boundary divisor $B$, consider a birational model $\pi:X' \to X$ and a boundary divisor $B'$ which is given by $K_{X'}+B'=\pi^*(K_X+B)$. Here is my ...
- 393
4
votes
0
answers
119
views
Are there any explicit (prime-to-l) alterations for interesting varieties (or schemes)?
I have read that it is easier to find regular alterations of varieties than their resolutions of singularities (moreover, I believed in this sentence when I read it). My question is: do there exist ...
- 16.9k
2
votes
1
answer
317
views
Singular locus of a linear system of hyperplane sections
Let $X\subset\mathbb{P}^N$ be a rational smooth projective irreducible non degenerated variety of dimension $n=\dim(X)$ and let $$\mathcal{H}=|\mathcal{O}_X(1) \otimes\mathcal{I}_{{p_1}^2,\dots,{p_l}^...
1
vote
0
answers
927
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 ...
- 623
1
vote
1
answer
216
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 ...
- 39.3k
2
votes
0
answers
163
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?
CommunityBot
- 1
2
votes
0
answers
61
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 $R^G$ ...
- 407
1
vote
0
answers
125
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}(...
- 4,189
1
vote
1
answer
160
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,...
- 1,306
4
votes
0
answers
320
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 ...
- 489
2
votes
1
answer
99
views
Uniformity of the set of poles of Igusa local zeta functions
Let $Ω_p$ denote the set of the real parts of the poles of the Igusa zeta function of a polynomial $f∈\mathbb{Z}[X_1,…,X_m]$ (assume $f(0)=0$ so that $\Omega_p\ne \emptyset$) at the prime p. From ...
3
votes
1
answer
190
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 ...
14
votes
0
answers
884
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, ...
CommunityBot
- 1
3
votes
0
answers
783
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 ...
- 5,966