All Questions
Tagged with resolution-of-singularities ag.algebraic-geometry
224 questions
5
votes
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
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. ...
- 1,981
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" ...
- 53
2
votes
1
answer
2k
views
Log Canonical pairs
Let $X$ be a normal scheme ad $D = \sum_id_iD_i\subset X$ be a $\mathbb{Q}$-divisor such thay $K_X+D$ is $\mathbb{Q}$-Cartier. Let $f:Y\rightarrow X$ be a log resolution of the pair $(X,D)$ and let us ...
user58018
5
votes
1
answer
1k
views
Hodge numbers of a Calabi-Yau 3-fold via deformation theory
In their paper "Calabi-Yau 3-folds and Moduli of Abelian Surfaces I", Gross and Popescu calculate the hodge numbers of smooth CY 3-folds obtained in the following way (see Remark 4.11 of their paper): ...
- 2,108
2
votes
0
answers
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
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 -\...
user47036
4
votes
2
answers
581
views
Singularities of Pfaffian hypersurfaces
Let $X\subset\mathbb{P}^4$ be an hypersurface of degree six given by the Pfaffian of a $6\times 6$ matrix $M$ whose entries are quadratic forms in the homogeneous coordinates of $\mathbb{P}^4$. I am ...
user61586
2
votes
0
answers
230
views
(Strict) canonical singularity with no crepant resolution
We know if a normal singular $\mathbb{Q}$-factorial variety $X$ has a crepant resolution, then it is canonical but not terminal. What can we say about the converse? To be more precise:
Is there a ...
- 815
7
votes
0
answers
177
views
How to compactify $\mathbb{Z}_p[x, y, z]/(xyz - p)$?
The affine scheme $U := \mathrm{Spec}(\mathbb{Z}_p[x_0, \ldots, x_d]/(x_0\cdots x_d - p))$ is regular and is a basic example of a semistable scheme over $\mathbb{Z}_p$. How does one build a proper ...
- 2,663
2
votes
0
answers
196
views
Total space of canonical bundle on toric del pezzo surface
Let $X$ be a toric del pezzo surface with a full cyclic strong exceptional collection of line bundles, say $\mathbb{E}:=(E_1,\ldots,E_n)$, consider the total space of anti-canonical bundle $\omega_X^*$...
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
2
votes
0
answers
271
views
Desingularization of subvariety
Let $X$ be a smooth projective complex variety. Let $Y \subset X$ is a singular closed subvariety of $X$. Does there exist a birational morphism $\pi: \widetilde{X} \to X$, such that the proper ...
- 569
3
votes
0
answers
157
views
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
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 ...
- 1,981
4
votes
2
answers
254
views
Invariant planes of a nilpotent matrix with two Jordan blocks of size two
Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map
$$
N = \begin{pmatrix}
0 & 1 & & \\
0 & 0 & & \\
& & 0 &...
- 1,123
3
votes
1
answer
647
views
Small resolutions are automatically crepant?
Page 17 of the following survey:
http://arxiv.org/abs/1103.5380
makes the claim that small resolutions, meaning resolutions such that the exceptional set is in codimension at least two, are ...
- 1,423
1
vote
1
answer
733
views
canonical divisors of a resolution of a normal surface singularity
Let $(0\in X)$ be the germ of a normal surface singularity and let $f: Y \to X$ be the minimal resolution.
Questions>
(1) How can I define a map $f_*\mathcal{O}_Y(K_Y)\hookrightarrow \mathcal{O}_X(...
- 53
3
votes
1
answer
268
views
Rationality of higher dimensional du Val singularities
I am interested in the isolated singularity defined over $\mathbb{C}$ by
$$
x_1^2+\cdots + x_n^2+x_{n+1}^k=0,
$$
where $n>2$ and $k>2$.
I would like to know whether this singularity is rational,...
- 2,384
2
votes
1
answer
221
views
Resolving nodes of a quintic CY 3-fold
Let's consider the following quintic 3-fold $X$:
\begin{equation}
\{(x_i) \in \mathbb{P}^4 \ | \ x_1f(x)-x_2g(x)=0\}
\end{equation}
for generic homogeneous polynomials $f(x),g(x)$ of degree four. It ...
- 21
4
votes
1
answer
619
views
Small resolution of a non-isolated singularity?
Consider the the Hypersurface singularity given by the equation
$$xyz+st=0 \subset \mathbb{C}^5.$$
How would you describe a (nice!=symmetric) small-resolution of this singularity?
1
vote
1
answer
1k
views
Singular irreducible quadrics
Let $Q\subset\mathbb{P}^n$ be the quadric hypersurface defined by
$$x_0^2+x_1^2+...+x_k^2 =0.$$
If $2\leq k\leq n-1$ then $Q$ is irreducible and $Sing(Q)$ is a linear space of dimension $n-k-1$.
If $...
user49214
3
votes
0
answers
289
views
Resolving quotient singularities without the quotient
This has been asked on MSE here, but has not had much traffic so I will ask a similar question here as many people here enjoy topics around resolving singularities.
Let $X$ be a complex threefold ...
- 362
1
vote
0
answers
179
views
Spivakovski-Popescu-Neron desingularisation
For $A \colon= {\Bbb F}_p[[X_1,...,X_d]]$, by generalising Popescu-Neron's method, Spivakovski proved that $A$ is written by smooth sub-algebras. That is,
$A \cong \underset{\lambda \in \Lambda}{\...
- 781
1
vote
0
answers
72
views
Minimal non-klt center of asymptotic linear system
Let $(X,\Delta)$ be a klt pair and $D $ a $Q $-Cartier divisor on $X $ such that the ring of sections of $D $ is finitely generated. Let $c$ be the log canonical threshold of the asymptotic linear ...
- 11
3
votes
0
answers
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
2
votes
1
answer
717
views
Singularities of secant varieties of rational normal curves
Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper:
http://ac.els-cdn.com/...
user56312
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
3
votes
0
answers
843
views
Projective tangent cones, ordinary singularities and blow-ups
Let $X\subset\mathbb{P}^n$ be a projective variety and let $Y\subset X$ be the singular locus of $X$. Assume that $Y$ is smooth. I would like to know if the following are equivalent:
$X$ has an ...
user61586
3
votes
1
answer
391
views
Embedded resolution of curves on smooth varieties
As far as I understand, embedded resolution of singularities means the following: given a variety $X$ over an algebraically closed field, and a closed subvariety $Y$, there exists a birational map $f:...
- 16.2k
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\...
- 2,218
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
2
votes
0
answers
191
views
Resolution of indeterminacies for a map to Grassmannian of planes
Let $X\subset \mathbb P_k^N$ be a $n$-dimensional smooth projective variety ($n\geq 2$) and $\phi_l:X^l\dashrightarrow Gr(l,N+1)$ ($l\leq n+1$) be the natural rational map which associates to a ...
- 155
0
votes
2
answers
490
views
Small birational maps and singularities of the pair
Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair $(X,\...
- 8,998
1
vote
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
-1
votes
1
answer
99
views
Finitely many subvarieties as divisor
Let $X$ be a smooth projective variety over an algebraically closed field of characteristic $0$ and of dimension $n\geq 10$. Let $(C_i)_{1\leq i\leq N}$ (resp. $(S_i)_{1\leq i\leq N}$) be smooth ...
- 1,463
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{...
- 1,981
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 ...
user47036
0
votes
1
answer
411
views
Intersection Matrix of a resolution
Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that
$$K_X = f^{*}K_S+\sum_ia_iE_i$$
with $a_i>0$. By Grauert-Mumford theorem the ...
user58018
2
votes
1
answer
202
views
Some Kind of Resolution of Singularites
Let $X \subseteq \mathbb{P}^n$ be a projective variety. I would like to have a morphism $f: \tilde{X}\to X \subseteq \mathbb{P}^n$ where $f$ is finite and birational, $f^* \mathcal{O}_{\mathbb{P}^n}(1)...
- 3,031
2
votes
0
answers
116
views
Constructive Resolution of Toric Singularities via Model Theory
Do there exists some language $\mathcal{L}$ of rational polyhedral cones in rational vector spaces and a theory $T$ over $\mathcal{L}$ whose models $\mathcal{M}$ are resolutions of toric singularities?...
- 1,595
1
vote
0
answers
183
views
Chern classes of a resolution of singularities
Let $j:X\subset \mathbb P_{\mathbb C}^n$ ($n\geq 3$) be a hypersurface, defined by a section of a very ample line bundle $\mathcal L$, with a ordinary double point $P$ as the only singularity and $\...
- 69
1
vote
0
answers
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
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) ...
- 44.7k
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 $E$ ...
- 2,108
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 ...
user47036
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 $...
- 81
1
vote
1
answer
687
views
A question on klt pairs
Let $D$ be a $\mathbb{Q}$-divisor in a smooth variety $X$. In Lazarsfeld book "Positivity in Algebraic Geometry 2" I found Proposition 9.5.13 saying that if for any $x\in D$ we have $mult_xD < 1$ ...
user58018
3
votes
0
answers
287
views
Does the invariant from resolution of singularities provide a Whitney stratification?
The topic of Whitney stratifications came up in a lecture, and the general procedure in the examples was to decompose the singular locus of the variety into the strata starting with the "worst" ones. ...
- 251
1
vote
0
answers
301
views
How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?
Let $X={\rm Spec} A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$.
Now,...
- 165