Questions tagged [resolution-of-singularities]
The resolution-of-singularities tag has no usage guidance.
245 questions
5
votes
1
answer
704
views
Simultaneous resolution of singularities in special cases of flat families of projective varieties
Let $\pi:\mathcal{X} \to B$ be a flat family of projective varieties. Assume that $B$ is irreducible. Suppose that $\mathcal{X}$ is smooth except for a closed subscheme, say $Y$ which is isomorphic to ...
- 4,111
1
vote
1
answer
156
views
Pushforward of $K_X+D$ on the non-snc locus
Let $f:Y\rightarrow X$ be a birational morphism of smooth projective varieties, $F$ an effective divisor on $X$, $D=f^{-1}F_{\mathrm{red}}+\mathrm{Ex}(f)$, $B$ a smooth subvariety of $Y$ contained in ...
- 4,111
4
votes
2
answers
1k
views
Varieties with big anti-canonical divisor
I recently heard about the following problem:
Let $X$ be a projective variety with klt singularities and such that $-K_X$ is big. Is $X$ a Mori Dream Space ?
Now, $-K_X$ big if and only if $-K_X -\...
- 8,998
7
votes
1
answer
341
views
How can one show that orbit closures in representations of a linear quiver don't have small resolutions?
Let $1\to \cdots\to n$ be a linear quiver of length $n$. Let $\mathbf{d}=(d_1,\dots,d_n)$ be a dimension vector. It's well known (for example, by Gabriel's theorem, but also by basic linear algebra) ...
8
votes
2
answers
741
views
Resolution of unpleasant singularity
I've been working on some varieties defined by taking some quotients of group actions, and the resolutions have been straightforward... until now.
E.g., consider $\mathbb{C}^2$ with the action $(x,y)\...
- 66.3k
4
votes
2
answers
1k
views
Crepant resolution of isolated fourfold singularity
I stumbled upon this isolated singularity of a Calabi-Yau fourfold:
\begin{equation}
x_1x_2+x_3x_4+x_5^2=0
\end{equation}
as a hypersurface in $\mathbb{C}^5$.
Clearly, I can resolve this by a simple ...
CommunityBot
- 1
2
votes
1
answer
386
views
Surfaces singular along a curve
Let $C\subset\mathbb{P}^3$ be a smooth curve a degree $d$ and genus $g$. Let $\mathcal{S}$ be the system of surfaces of degree $k$ in $\mathbb{P}^3$ containing $C$ with multiplicity $\beta$.
What is ...
- 38k
3
votes
2
answers
513
views
Which isolated surface singularity comes from a -5 curve?
Define the surface $X$ to be the total space of $\mathcal{O}_{\mathbb{P}^1}(-5)$.
By contracting the exceptional curve in $X$, we get a surface with an isolated singularity. I am looking for the ...
- 8,998
0
votes
1
answer
700
views
Kawamata-Log-Terminal pairs
Let $p_1,...,p_n\in\mathbb{P}^3$ be general points, and let $\Delta\subset\mathbb{P}^3$ be a general surface of degree $d$ with points of multiplicity $m_i$ at $p_i$ for $i = 1,...,n$.
Consider the ...
CommunityBot
- 1
3
votes
1
answer
957
views
Is it possible to resolve singularities using only normal varieties?
In characteristic 0, is it possible to have a resolution of singularities where the algebraic varieties at every step of the desingularization process are normal. To be more precise, I would like a ...
- 251
2
votes
1
answer
110
views
Determining the desingularization from the complete local ring
Suppose I have a curve $C$ over a field $k$ and that $p$ is a singular point of $C$. Let $f : X \to C$ be the desingularization of $C$ at $p$. Then for each $s \in f^{-1}(p)$ we have a map of local ...
- 38k
4
votes
1
answer
399
views
Blowing up rational singularities
Let $X$ be a projective surface embedded into $\mathbb{P}^n_{\mathbb{C}}$ having at most rational singularities. Let $\tilde{X} \to X$ be the minimal resolution of $X$. Is it possible to embed $\tilde{...
- 38k
3
votes
0
answers
287
views
big and small resolutions of singularities of a 4-fold
Suppose we have a projective 4fold hypersurface $X\subset P^n$
with ordinary singularities along a smooth curve $C$, and suppose that there exist a projective small resolution $s:Y\to X$. let us ...
- 3,779
3
votes
3
answers
392
views
Contractibility of curves and embedding into projective space
Let $f:X \to Y$ be a proper surjective morphism of projective surfaces such that there exists a curve $C \subset X$ for which $f|_{X\backslash C}$ is an isomorphism and $f(C)$ is a set of points. ...
- 42.9k
2
votes
1
answer
189
views
minimality/universality of the Springer resolution of a determinantal variety
Let $X\subset P^n$ be a singular determinantal variety and $S\to X$ its Springer resolution. Let $X'\to X$ another resolution of singularities (say, a blow-up). Does $S$ have some minimality/...
- 7,320
9
votes
2
answers
622
views
Can we foliate the punctured space by tori?
Is it possible to have a 2 dimensional foliation of $\mathbb{R}^{3}-\{0\}$ such that each leaf is homeomorphic to the torus? what algebraic topological obstruction exist?
Another question: is there ...
- 366
3
votes
0
answers
272
views
References for resolutions of ordinary singular points
Let $X$ be a $n$-dimensional complex projective algebraic variety, let us suppose that $X$ has only isolated singularities.
Edit: Let us say that an ordinary $m$-ple singular point is an isolated ...
- 9,870
4
votes
1
answer
619
views
Small resolution of a non-isolated singularity?
Consider the the Hypersurface singularity given by the equation
$$xyz+st=0 \subset \mathbb{C}^5.$$
How would you describe a (nice!=symmetric) small-resolution of this singularity?
- 126
1
vote
1
answer
471
views
Serre's conditions under blow-ups, Blowup and normalization
Suppose $X = \mathbb{Z}[x, y, z]/(f,g)$ is a 2-dimensional Cohen-Macaulay surface. In particular, $X$ satisfies Serre's condition $S_2$. Suppose it is irreducible, reduced but not normal.
$\bf{...
- 20.5k
2
votes
0
answers
137
views
Global minimal model over a non-affine base
Remark 10.1.8 in Liu's book (AG and Arithmetic curves) says that over a non-affine base (base is always assumed to be a Dedekind scheme of dim 1), the minimal regular model of a (smooth projective) ...
- 6,210
9
votes
2
answers
870
views
Finite generation and Henselization
Now I understand the answers.
I am trying to understand Henselian Weierstrass Theorem in Hironaka's Idealistic exponents of singularity, page 76 - 77. A glance at the paper.
At some point there is ...
- 807
3
votes
0
answers
531
views
"Step-by-Step" toric resolution process?
WLOG the fan $\Sigma$ of our toric variety $X_{\Sigma}$ is simplicial. (So $X_{\Sigma}$ has at worst orbifold singularities and all cones $\sigma \in \Sigma$ are simplicial).
The classical toric ...
- 1,939
1
vote
1
answer
715
views
Importance of Denjoy-Carleman classes as a class.
Denjoy-Carleman classes of differentiable functions, say in Roumieu's form:
Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of $C^\infty(\...
- 25.3k
3
votes
0
answers
175
views
(semi-)Small resolutions of Peterson varieties
Peterson varieties (in type A) can be described as the subvarieties of the full flag variety
$$\{(F_{i})\;|\; F_{i}\subset \mathbb{C}^{n}, \; \dim F_{i} =i,\; N(F_{i})\subset F_{i+1}\}$$
where $N$ ...
- 1,201
0
votes
0
answers
109
views
solve the singularities of parabolic orbits of schubert cells
Let G a semsisimple connect'ed group over $k$, $B$ a Borel and $P$ a parabolic subgroup of $G$ with Weyl group W_{P}.
For $w\in W_{P}\backslash W/W_{P}$, how can we solve the singularities of $X_{w}=\...
- 3,472
3
votes
1
answer
209
views
Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite?
Let $X$ be a smooth complex manifold with finite fundamental group. Suppose that a finite group $G$ acts on $X$ and let $\widetilde{X/G}$ be a resolution of singularities. Is $\pi_1(\widetilde{X/G})$ ...
CommunityBot
- 1
4
votes
1
answer
537
views
Which schemes can be presented as limits of smooth varieties?
I can prove a certain statement for any scheme that can be presented as the limit of an essentially affine (filtering) projective system of smooth varieties over a perfect field such the connecting ...
- 16.9k
14
votes
0
answers
857
views
Is a flop on Calabi-Yau threefolds always Atiyah flop?
Is it true that any flop on a Calabi-Yau threefold is given by the Atiyah flop? That is, there always exists a rigid rational curve $\mathbb{P}^1$ with normal bundle $\mathcal{O}_{\mathbb{P}^1}(-1)^{\...
- 427
4
votes
1
answer
494
views
Normal bundle of exceptional locus of the conifold
Let us consider the conifold singularity $xy-zw=0$ in $\mathbb{C}^4$. By blowing up along the divisor defined by $x=z=0$, we have a small resolution of the conifold with $\mathbb{P}^1$ as the ...
- 39.3k
5
votes
2
answers
3k
views
Embedded resolution of singularities
I'd like to check with my colleagues whether I have correctly understood "embedded resolution of singularities".
Let $X$ be a nonsingular projective variety over $\mathbf C$ and let $D$ be a "nice" ...
- 42.9k
2
votes
1
answer
281
views
resolution of strata of the affine grassmanian
Let G a semisimple simply connected group over an algebraically closed field.
Let $Gr:= G(k((t))/G(k[[t]])$ be the affine grassmanian. It admits a stratification indexed by the dominant cocaracter
$...
- 27.9k
5
votes
2
answers
694
views
Crepant resolutions of ODP's on a 3-fold
It seems to be a well-known fact that if we have a 3-fold $X$ with only ODP singularities (ordinary double point) and a smooth Weil divisor $E$ passing through them, then by blowing-up $X$ along $E$ ...
- 42.9k
2
votes
2
answers
353
views
springer resolution over $\wedge^3 \mathbb{C}^6$
The action of $GL_6$ on $P(\wedge^3 \mathbb{C}^6)=P^{19}$ has 4 orbits (of dim 19, 18, 14, 9). Can you describe how the springer resolution applies to each of these orbits? It should have positive ...
- 39.3k
8
votes
1
answer
773
views
Schubert varieties which admit small resolutions of singularities
I am looking for an (incomplete) list of partial flag varieties for
which all Schubert cells admit small resolutions of singularities.
This is interesting, for many reasons. My motivation is, that a ...
- 3,103
8
votes
1
answer
978
views
When do blow-up and quotient commute?
Let a finite group $G\subset SL(n,\mathbb{C})$ act on $\mathbb{C}^{n}$ in a natural way. Assume there is a crepant resolution of $f:X\rightarrow \mathbb{C}^{n}/G$. When is it possible to write $X$ as $...
- 39.3k
3
votes
1
answer
340
views
2
votes
0
answers
143
views
compatible resolutions of singularities
Let $U$ and $V$ be complex vector spaces with an action of a finite group $G$. Denote by $P_{G,d}$ the space of $G$-equivariant polynomial maps $U\longrightarrow V$ with degree less than or equal to $...
- 483
2
votes
2
answers
506
views
Line bundles and rational singularities
Hi, I have some problem to understand the proof of lemma 3.2 of this article: http://www.ams.org/journals/jams/2001-14-03/S0894-0347-01-00368-X/.
The lemma states the following:
Let $X$ be a variety ...
- 66.3k
4
votes
0
answers
487
views
Embedding of a smooth variety into a complete smooth variety.
Consider the following fact from algebraic geometry:
Any (complex) smooth algebraic variety can be embedded into a complete smooth variety as a locally closed set.
I know how to prove this fact ...
- 2,649
2
votes
2
answers
269
views
commuting the resolution of 1-dim singular locus and 0-dim singularities in a non isolated singularity of a surface
Let $X$ be a surface with a non isolated singularity $C = Sing(X)$ such that the curve $C$ has singularities itself. We can solve $Sing(X)$ by blowing up close points and by normalizing. Indeed, we ...
- 20.5k
1
vote
1
answer
573
views
'Reference' request: Program to work with cyclic quotient singularities.
I'm looking for program code to deal with cyclic quotient singularities on normal surfaces. In particular, at the moment I need that given a singularity $p$ like $p=\frac{1}{n}(1,a)$ the code computes:...
- 3,625
4
votes
2
answers
611
views
Vanishing associated to a resolution of singularities
Let $\pi: V\to W$ be a resolution of singularities, let $E \subset V$ be the exceptional divisor, and let $F$ be a coherent sheaf such that $R^i\pi_*F=0$ for $i>0$.
Can we conclude that $R^i\...
- 42.9k
10
votes
2
answers
1k
views
Resolution of singularities for flat families.
Is there a resolution of singularities for flat families?
More precisely, if $X \rightarrow \mathbb{A} ^n$ is a flat map, does there exist a map $Y \rightarrow X$ such that, for every $p \in \mathbb{...
- 42.9k
4
votes
1
answer
363
views
Property of singularity
Let $X$ be an algebraic variety, $S \subset X$ its singular locus, and $x \in S$. Say that $x$ is good if for any neighborhood $U$ of $x$, any top differential form $\omega$ on $U \setminus S$ and ...
- 20.5k
14
votes
0
answers
547
views
When are the fibers of a resolution of singularities reduced?
I apologize if this is too much of a fishing expedition, but I've had bad luck searching for any literature on this subject, and I was hoping someone could tell me if it's too easy to be worth ...
- 44.7k