Questions tagged [blow-ups]
The blow-ups tag has no usage guidance.
185
questions
2
votes
0
answers
36
views
0-framed smoothly slice knot that can be obtained by blowing down successively a link of unknots
A knot in $S^3$ is called a smoothly slice knot if it bounds a smoothly embedded 2-disk in $D^4$. Every ribbon knot is known to be a smoothly slice knot, and there are known some nontrivial smoothly ...
- 161
0
votes
0
answers
123
views
Vector bundles on blowups
$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\ker{ker}$Let $X$ be a (projective) variety, which is singular at a point $p$ of $X$ (we can also replace $p$ with some closed subvariety). Suppose $Y$...
- 422
0
votes
0
answers
100
views
Nonlinear wave equation : ODE blow-up construction
Consider the focusing non-linear wave equation (NLW)
\begin{align*}\partial_{tt} u(t,x) - \Delta u(t,x) &= -|u|^{p-1}u, \quad (t,x) \in [0,+\infty) \times \mathbb R^d \\
u(0,x) &= u_0(x) \in \...
- 101
0
votes
1
answer
130
views
Blow up and critical points of the projection map
Denote $Z=V(x_1, \dots, x_{n-1}) \subset \mathbb{C}^n$ and let $Bl_Z(\mathbb{C}^n)$ be the blow up of $\mathbb{C}^n$ along $Z$ together with the projection map $\pi \colon Bl_Z(\mathbb{C}^n) \to \...
2
votes
0
answers
92
views
On intersections of exceptional divisors
Let $X$ be a smooth, projective variety of dimension $n \ge 2$, $L$ a very ample line bundle on $X$ and $\pi: \widetilde{X} \to X$ be the blow-up along a closed subvariety of codimension at least $2$. ...
- 1,940
4
votes
0
answers
178
views
Blow-up of a stratified space
Let $X$ be a smooth projective variety over $\mathbb{C}$, and $D_1, \ldots, D_n$ be a collection of simple normal crossing divisors. The divisors induce a stratification $\mathcal{T}_X$ of $X$.
Let $...
- 223
1
vote
0
answers
80
views
Representability of blow-up of stacks
Suppose $\mathcal{Z}\subseteq\mathcal{X}$ is a closed immersion of stacks. Is the blow-up morphism $p:Bl_{\mathcal{Z}}(\mathcal{X})\to\mathcal{X}$ representable? More precisely, if $C\to\mathcal{X}$ ...
- 710
6
votes
1
answer
387
views
If power of an ideal is locally free then it is locally free
Let $X$ be a noetherian scheme and $\mathcal{I} \subset \mathcal{O}_X$ a coherent sheaf of ideals. Suppose that $\mathcal{I}^d$ is locally-free for some power $d$. Then the blowing up $\mathrm{Bl}_{\...
- 1,822
0
votes
0
answers
104
views
Problem in calculating the global sections of $\mathcal{O}_{\mathbb{P}^3}(d)\otimes \mathcal{I}_Z$
This is an additional question to the one I posed in Equivalence of sequences of blowups of $\mathbb{P}^3$
Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the ...
- 1,167
2
votes
1
answer
184
views
Equivalence of sequences of blowups of $\mathbb{P}^3$
Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the subscheme given by the ideal $$I_Z=(x_1,x_2,x_3^2) \subset \mathbb{C}[x_1,x_2,x_3,x_4]$$ i.e. $Z$ is a double ...
- 1,167
2
votes
1
answer
134
views
Mori fiber space contractions of the blow-up of a projective space
Let $\mathbb P(V)$ be a projective space containing $Y$ as a subvariety. Let $Z$ be the blow-up of $\mathbb P(V)$ along $Y$.
Clearly there exists a divisorial contraction given by $b: Z \to \mathbb P(...
- 291
9
votes
1
answer
677
views
Inverse Kirby knot
Given an (oriented framed) knot $K$ in the 3-sphere $S^3$, we can perform a surgery along $K$ to get another 3-manifold $M$. From $M$, we can perform the inverse surgery back to $S^3$.
However, the ...
- 4,005
9
votes
0
answers
265
views
Proof of Artin–Rees / Krull intersection motivated by universal property of blowup
I was very confused by the proof of Artin–Rees / Krull intersection theorem when I was younger.
Now that I learnt about blow up— I saw the Rees algebra again and I want to now gain a better ...
- 485
3
votes
0
answers
119
views
How to use blow-up to prove the boundary regularity for a harmonic function
While reading the book Regularity Theory of Elliptic PDE I’m confused with a theorem:
Thm. 2.30.
Let $\alpha \in (0,1)$ and $k \in N$ with $k \leq 2$, and let $\Omega$ be a bounded $C^{k, \alpha}$ ...
- 87
2
votes
0
answers
157
views
On the exceptional divisor of the resolution of indeterminacy locus of rational map
Let $f:X \dashrightarrow Y$ be a rational map between smooth, projective varieties over $\mathbb{C}$. We know that there is a resolution of the indeterminacy locus using which we obtain a smooth, ...
- 1,940
2
votes
1
answer
170
views
Blow-ups of surfaces over a field
Let $S$ be a smooth projective surface of Picard rank $\rho(S)$ over a field $K$, and $\overline{S}$ its algebraic closure.
Take a point $p\in\overline{S}$ and denote by $\overline{X}$ be blow-up of $\...
user329228
5
votes
1
answer
218
views
Which projective varieties cannot appear as an exceptional divisor of a blow-up of the projective space
I am looking for examples of projective varieties (over $\mathbb{C}$) of dimension, say $n$ which cannot appear as an exceptional divisor of a blow-up of $\mathbb{P}^{n+1}$ along some closed subscheme....
- 2,046
2
votes
0
answers
138
views
Base locus of a linear system after blow-up
I have a smooth proejective fourfold $X$ with an effective divisor $D$ on it.
The base locus ${\rm Bs}|D|$ of the linear system $|D|$ is a smooth rational curve $C$ and a generic member of $|D|$ has ...
- 1,051
10
votes
1
answer
662
views
What's the cohomology ring structure of a blow-up?
Let $X$ be a compact Kähler manifold, with $j_Z: Z\hookrightarrow X$ a submanifold of complex codimension $r$, $\tau: \widetilde{X} \to X$ the blow-up of $X$ along $Z$, with exceptional divisor $j: E \...
- 368
2
votes
0
answers
147
views
Higher cohomology of projective bundles
Let $C$ be a curve and $L$ be a line bundle with sufficiently large degree. Let $C_p$ denote the $p$-th symmetric product of $C$, which consists of all the effective divisors of degree $p$ on $C$. Let ...
- 305
2
votes
0
answers
130
views
Intersections of strict transform and strict transform of intersections
Let $Z_1,Z_2$ and $Y$ be subvarieties of a locally complete intersection variety $X$ over $\mathbb C$.
Consider the strict transforms of $Z_1$ and $Z_2$ in the blowup $Bl_YX$, the question is: when ...
- 21
2
votes
1
answer
120
views
Is the blowup of a toric variety corresponding to a subdivision normal?
Toroidal Embeddings 1 by KKMS say subdividing the fan of a toric variety yields the fan of a normalized blowup. How do I avoid normalization? Do I need to choose my subdivision carefully, or is it ...
- 924
2
votes
0
answers
179
views
Surgery for algebraic varieties
I have a number of vague questions that I wasn't sure whether they are suitable to ask or not, but I decided to ask!
According to this result, any two birational varieties can be constructed by a ...
- 4,926
1
vote
0
answers
128
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 ...
- 4,926
2
votes
1
answer
177
views
Blow-up of a three-dimensional variety at a node
Let $X$ be a three-dimensional variety over $\mathbb{C}$ with a nodal singularity at a point, say $P$. Is the exceptional divisor of the blow-up of $X$ at $P$ isomorphic to a smooth quadric in $\...
- 1,940
2
votes
1
answer
225
views
Help about "Varieties with small Dual Varieties" by L.Ein
I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding ...
- 1,167
9
votes
5
answers
771
views
Blow-up of projective variety $P^1 \times P^1..... \times P^1$ ($n$ times) and blow-up of $P^n$
It is known that the blow-up of $P^1 \times P^1$ at a point is isomorphic to the blow-up of $P^2$ at two points. I'm wondering if there is any general statement for the blow-up of $P^1 \times P^1 \...
- 177
7
votes
1
answer
310
views
Strict transform of a tangent curve under blow-up
$\DeclareMathOperator{\Bl}{\operatorname{Bl}}$It is known that if we have a projective variety $X$ and a projective smooth subvariety $Y$ then the exceptional divisor $E \subset \Bl_{Y}X$ of the blow-...
- 1,167
3
votes
0
answers
123
views
branch divisor of this map
We consider the blow up $Bl(\mathbb{P}^2)_p$ of $\mathbb{P}^2$ in $p:=|1:0:0|$ and the following surface:
$Y:=\{(|y_1: y_2:y_3:y_4|, |x_0:x_1:x_2|) \in \mathbb{P}^3\times \mathbb{P}^2: rk(\begin{...
7
votes
1
answer
348
views
General conditions for normality of blow-up
Let $X$ be an integral, affine, normal complex surface. I am looking for conditions on zero-dimensional closed subschemes $Z$ in $X$ such that the reduced scheme associated to the blow-up of $X$ along ...
- 1,940
4
votes
1
answer
326
views
Relative canonical class of blowing-up a flag ideal
Let $X$ be a smooth complex projective variety of dimension $n$. Consider a flag ideal $I$ on $X\times \mathbb{P}^1$, namely,
$$
I=I_0+I_1t+I_2t^2+\cdots+I_{N-1}t^{N-1}+(t^N)\,,
$$
where $t$ is the ...
- 159
4
votes
0
answers
85
views
Serre vanishing on one-point blow-ups
This is basically the last step of problem 5.3.7 in Huybrechts' Complex Geometry.
Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. ...
- 527
5
votes
1
answer
176
views
Extending rational maps of nodal curves
Let $R$ be a discrete valuation ring with fraction field $K$ and $C, D$ two nodal (=prestable) curves over $\operatorname{Spec} R$. If I have a map $C_K \to D_K$ between the restriction of the curves ...
- 924
5
votes
0
answers
224
views
Deformations of a blow up
My question is related to this question, but I'm looking for something a bit more explicit.
Let $S$ be a smooth surface over $\mathbb C$, fix a point $s\in S$ and take the blow up $\beta \colon S' \...
2
votes
0
answers
113
views
Determining if a morphism is a blowup along a given subvariety
Let $X,\tilde{X}$ be two smooth projective varieties over $\mathbb{C}$, and let $\pi:\tilde{X}\rightarrow X$ be a projective morphism. Let us moreover assume that there exists a smooth closed ...
- 533
1
vote
0
answers
76
views
Properties of the contraction map of the exceptional divisor of a blow-up
Let $X$ be a smooth variety and $Z$ be a smooth subvariety. Consider the blow-up at $Z$ $$\pi:\mathrm{Bl}_Z(X)\rightarrow X$$ and let $E$ be the exceptional divisor. What are the properties of the ...
- 291
2
votes
1
answer
240
views
Picard group modulo codimension 2
Let $X$ be a normal (possibly singular) projective surface over $\mathbb{C}$. Consider the set $M_X$ of all coherent sheaves $F$ on $X$ such that there exists a finite subset $Y\subset X$ such that $F$...
- 2,277
6
votes
2
answers
406
views
Simple proof that the arithmetic genus is non-negative
I take an irreducible and reduced closed curve $C\subseteq \mathbb{P}^n$, defined over an algebraically closed field $k$ and define the arithmetic genus $p_a(C)$ as the integer such that the Hilbert ...
- 7,180
5
votes
0
answers
165
views
When do the spectra of overrings glue to a proper morphism?
This question is motivated by the construction of blowups.
Let $A \subset K$ be a commutative domain and its fraction field, and let $\{A_i\}$ be some finite collection of overrings in between.
Let $X ...
- 1,295
1
vote
1
answer
99
views
Reference requence: scheme of complete homomorphisms of rank $r$ via blowups
I'm reading these notes
where it states in section $3$: (transcribed because I can't post image)
Step 1. Introduce the stacks of degenerated and iterated shtukas
which extends that of shtukas.
This ...
- 113
4
votes
1
answer
466
views
Blowing up vector bundles in the zero section
Assume we are given a scheme $X$ (feel free to add all the needed hypotheses, at this point I’m working with smooth schemes, but the fewer is needed, the better) and a vector bundle $E$ over $X$. I ...
- 1,295
1
vote
0
answers
212
views
Extend any morphism to suitable projective variety? [closed]
Let $F: X\to \mathbb{P}^n$ be a morphism from an affine variety to projective space (over some algebraically closed field of characteristic zero). Can we always find an open immersion $\iota: X\...
- 2,277
5
votes
0
answers
251
views
Fibers of blow up in families
Let $T$ be a smooth curve over $\mathbb{C}$ and $p:\mathbb{P}^n \times T \to T$ the natural projection. Let $V \subset \mathbb{P}^n_T$ be a $T$-flat subscheme of codimension at least $2$ and $\pi: \...
- 1,939
10
votes
0
answers
170
views
Derived equivalences preserved by blow-ups
Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. Assume that $X$ and $Y$ are derived equivalent. Let $\pi : \tilde{X} \longrightarrow X$ be a blow-up of $X$ along a smooth center. Can ...
- 6,159
8
votes
2
answers
534
views
Blow-up of the plane at $5$ points
If we Blow-up $\mathbb P^2_{\mathbb C}$ at $5$ points $T=\{p_1,\ldots,p_5\}$ we obtain a Del Pezzo surface $X$ of degree $4$. Now take another set of $5$ points $T'=\{q_1,\ldots,q_5\}$ ($T'\neq T$), ...
- 529
0
votes
0
answers
94
views
Proper subvarieties of blow-up
Let $X$ be an integral affine scheme of finite type over $\mathbb{C}$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get a scheme $Bl_Y X$. ...
user74900
7
votes
1
answer
807
views
Why only some del Pezzo are toric?
Let us define smooth del Pezzo surfaces $dP_r$ as the blowup of $r$ generic points in $\mathbb{CP}_2$. One can show that if we request $dP_r$ to be Fano, then $r=0,...,8$. In theoretical physics ...
- 469
1
vote
0
answers
75
views
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 ...
- 355
8
votes
1
answer
357
views
The relative dimension of blow-up and singularities
Let $X$ be an integral affine scheme of finite type over $\mathbb{C}$, $\mathrm{dim}\,X=d$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get ...
user136894
1
vote
1
answer
163
views
Projective subvarieties of blow-ups of affine varieties
Let $X$ be an integral affine scheme of finite type over a field $k$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get a $k$-scheme $X'$. Is ...
- 231