All Questions
Tagged with resolution-of-singularities singularity-theory
61 questions
1
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0
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57
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Discrepancy of general element of linear system
Let $X$ be a normal scheme and $|D|$ a linear system on $X$.
In "Singularity of Minimal Model Program" by Janos kollar p249, it says,
If $X$ is a variety over $\mathbb{C}$, and $E_j$ ...
- 328
2
votes
1
answer
117
views
Blow up of terminal singularity and canonical singularity
A normal singularity $(X,x)$ over a field $k$ is terminal (resp. canonical) if
$(i)$
it it is $\mathbb{Q}$-Gorenstein. and
$(ii)$For any resolution of singularity $F:Y\rightarrow X$,
$K_Y-f^*K_X>...
- 328
1
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0
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62
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About the definition of cDV singularity
M. Reid defines cDV singularity as follow in his paper "CANONICAL 3-FOLDS"
A point $p\in X$ of a 3-fold is called a compound Du Val point if for some section H throgh $P$, $P\in H$ is a Du ...
- 328
2
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0
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108
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Finiteness of rational double point
Let $(R,\mathfrak{m
})$ be a three dimensional complete local ring over a field $k$ of arbitrary characteristic and let $f\in R$ and $R/f$ is a rational double point.
My question is as follows.
Are ...
- 328
1
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1
answer
151
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Examples of small resolutions in dimension 4 and higher
I have seen numerous examples of 3-folds admitting small resolutions. Are there similar examples in higher dimension? In particular, I am looking for examples of singular varieties of dimension $4$ ...
- 2,323
2
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1
answer
327
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Krull dimension of the smooth locus
Let $R$ be a normal complete local domain of dimension $n \geq 4$. Does there exist a prime ideal $\mathfrak{p}$ of height $\dim(R) - 1$ such that $R_{\mathfrak{p}}$ is a regular local ring? In ...
2
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1
answer
304
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Normal forms of ADE singularities
Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms?
Does a similar ...
- 481
0
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0
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83
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How to determine the singlarity type (up to local analytic isomorphism) of a hypersurface surface singularity
Given a polynomial f(x,y,z), it defines a hypersurface in $\mathbb C^3$. I guess there is a classification of hypersurface singularity like Arnold normal form. I wonder given an explicite example of f,...
- 623
2
votes
0
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108
views
Deformation to normal cone of the exception divisor of a log-resolution
I am reading the paper Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink due to G. Guibert, F. Loeser, and M. Merle. The main tool, like a lot of papers in ...
- 893
1
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1
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209
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Induced resolution of singularities
I am not a specialist in singularity theory but currently I have to touch resolution of singularities and I'd like to know whether I have understood Hironaka's theorem correctly.
Let $k$ be a field of ...
- 893
2
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0
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203
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Trace formula for monodromy of Milnor fibrations
I am reading the paper A. Campo, Le nombre de Lefschetz d'une monodromie but I am stuck at several points, hope that someone can help me.
Let $P:\mathbb{C}^{n+1} \longrightarrow \mathbb{C}$ be a germ ...
- 893
5
votes
1
answer
410
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Understand the proof that rational resolution is independent of the resolution
EDIT. Karl Schwede has given a different approach to solve this question, which is very clear. However, I still want to figure out how to deal with this problem along Ein's line, and waiting for the ...
- 295
4
votes
1
answer
387
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Do there exist linear relations between exceptional divisors
Let $X$ be an isolated, Gorenstein singularity of dimension at least $2$ and $\pi: \widetilde{X} \to X$ be a resolution of singularities. Let $E$ be the exceptional divisor and $E_1,...,E_r$ be the ...
- 2,032
2
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1
answer
238
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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 ...
- 151
2
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0
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220
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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
8
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1
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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
5
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1
answer
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
6
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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
3
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0
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992
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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
6
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0
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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
7
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1
answer
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
2
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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
1
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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
0
votes
0
answers
82
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
1
answer
383
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 ...
- 623
4
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0
answers
116
views
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
answers
451
views
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
1
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40
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On Remmerts reduction
Let $(X,0)$ be a normal surface singularity. An let $\pi: \tilde{X} \to X$ be the minimal resolution. Now, we can apply a result of Oliveira (exploiting previous work by Laufer) and obtain a 1-...
- 1,409
8
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1
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496
views
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
1
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0
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219
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Desingularization of the zero section of $TM$ as the manifold of singularities of the geodesic flow
However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities"...
- 366
4
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0
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168
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Can nonflat deformations of singularities always produce Cohen-Macaulay rings?
To make the question in the title precise, let me phrase it like this. Consider a complete local ring
$$ A := \mathbb{C}[[x_1, \dotsc, x_n]]/(f_1, \dotsc, f_m) $$
and, for definiteness, assume that $...
- 2,663
2
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0
answers
149
views
Reference for certain resolution of singularities formulation
I want to use the following resolution of singularity statment as found in Soule et al, Lectures on Arakelov Geometry, p. 40:
$Y$ is a separated algebraic variety of finite type over $\mathbb{C}$, $Z$...
- 195
2
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1
answer
399
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resolution for the du Val's $(A_3)$-singularity
For the $A_m$-singularity, it can be viewed as the singular part of $\mathbb{C}^2/\mathbb{Z}_m$. The action of $\mathbb{Z}_m$ on $\mathbb{C}^2$ is defined as following
$$
\bar{1} \cdot (z,w) = (z e^{\...
- 167
1
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0
answers
328
views
Cohomology of a structure sheaf of a normal affine variety
I can't find the reference for the following fact:
Let $X$ be an affine variety and let $Y$ be its smooth resolution. $H^0(X,\mathcal{O}_x)=H^0(Y,\mathcal{O}_Y)$ if and only if $X$ is normal.
- 371
4
votes
1
answer
330
views
example of quintics with 5 ordinary triple point
I know we can bound the triple point on quintics in cp^3 by 5. But how to write down quintics with 5 ordinary triple point (here are simple elliptic singularity)explicitly?
- 623
9
votes
1
answer
1k
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Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?
Let $X$ be an affine variety. Let $Y$ be smooth and let the map $f\colon Y\rightarrow X$ be proper birational. We will call $Y$ a smooth resolution of $X$.
Do the cohomology groups $H^i(Y,\mathcal{O}...
- 371
3
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0
answers
352
views
smoothing of isolated surface singularity
I want to know when an isolated surface singularity can be smoothed, especially for log canonical isolated surface singularity. Is there any good reference. Thanks in advance.
- 623
93
votes
0
answers
17k
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Hironaka's proof of resolution of singularities in positive characteristics
Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier...
Unlike Atiyah's paper, Hironaka's paper does not have a ...
- 8,071
3
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0
answers
447
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simple elliptic surface singularity
Suppose X is a one dimension torus and L is a line bundle over X, I think one class of log canonical surface singularity comes from the contraction of one elliptic curve from the total space L. My ...
- 623
2
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0
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214
views
log canonical surface singularity
For a log cannonical surface singularity, I guess there is classification about the configuration of the exceptional divisor of its minimal resolution. I wonder if there is some specific example or ...
- 623
3
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0
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82
views
Singularities of fibrations 2
This question is related to my previous question:
Singularities of fibrations
Assume that $X$ is a complete intersection irreducible $3$-fold in a product of projective spaces. So that $X$ is ...
- 8,998
4
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1
answer
327
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Singularities of fibrations
Let $f:X\rightarrow \mathbb{P}^2$ be a fibration, here $X$ is a projective variety of dimension three.
Assume that there exixts a smooth curve $C\subset\mathbb{P}^2$ such that for any $p\in\mathbb{P}...
- 8,998
2
votes
1
answer
167
views
Singularities of $3$-folds
Let $X,Y,Z$ be projective $3$-folds. Assume that $Y$ is smooth and $Z$ is smooth and Fano. Moreover, assume that there is a generically finite morphism $f:Y\rightarrow Z$ admitting a factorization $f=...
user97096
5
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0
answers
319
views
Resolving analytic normal crossings singularities
Let $X$ be a non-singular (complex) variety and $Y \subset X$ be a (reduced) irreducible subvariety with only normal crossings singularity (locally, in the analytic topology, the singularity is ...
- 1,981
3
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0
answers
157
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Resolving complete intersections of quadrics with singularities
Suppose that $X$ is a complete intersection of quadrics in $\mathbb P^n_{\mathbb C}$. Is there some straightforward procedure to resolve the singularities of $X$?
For example, can one stratify ...
- 14.3k
1
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0
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74
views
Simple question about surface singularities
Given $\epsilon \in (0,1)$, is it possible to find two finite familes $\mathcal{F}$ and $\mathcal{P}$ of weighted graphs, such that the weighted graph of the minimum resolution of any $\epsilon$-klt ...
- 1,595
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 ...
user91659
1
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0
answers
189
views
A definition of arithmetic divisor with conic singularities?
I have a question related to the preprint "Heights and metrics with logarithmic singularities" by G. Freixas i Montplet.
Let $X$ be an arithmetic variety with arithmetic divisor $D$ how can we ...
user21574
3
votes
1
answer
268
views
Rationality of higher dimensional du Val singularities
I am interested in the isolated singularity defined over $\mathbb{C}$ by
$$
x_1^2+\cdots + x_n^2+x_{n+1}^k=0,
$$
where $n>2$ and $k>2$.
I would like to know whether this singularity is rational,...
- 2,384
8
votes
3
answers
918
views
Contracting a rational curve in a Calabi-Yau threefold
Let $X$ be a Calabi-Yau threefold and $C \subset X$ be a rational curve with $N_{C/X}\cong \mathcal{O}\oplus \mathcal{O}(-2)$. Can one contract the curve $C$? Assuming the answer is yes, what kind of ...
- 81