Questions tagged [blow-ups]
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77
questions with no upvoted or accepted answers
17
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0
answers
820
views
Horrible sets and blowups in Hubbard's Teichmuller theory
Edit: I can rephrase this question this way: When blowing up every point in the $x$-axis in $\mathbb{C}^2$ by means of an inverse limit of finite blowups, how can anything be 'left over'? The horrible ...
- 3,309
13
votes
0
answers
530
views
Cohomology of a blow-up of a real algebraic variety
Let $X$ be a complex algebraic variety, $Z \subset X$ a closed subvariety, $\mathrm{Bl}_Z X$ the blow-up and $E$ the exceptional divisor. There is an isomorphism of cohomology groups
$$ H^k(X(\mathbf ...
- 39.3k
10
votes
0
answers
182
views
Derived equivalences preserved by blow-ups
Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. Assume that $X$ and $Y$ are derived equivalent. Let $\pi : \tilde{X} \longrightarrow X$ be a blow-up of $X$ along a smooth center. Can ...
- 7,200
9
votes
0
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333
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Proof of Artin–Rees / Krull intersection motivated by universal property of blowup
I was very confused by the proof of Artin–Rees / Krull intersection theorem when I was younger.
Now that I learnt about blow up— I saw the Rees algebra again and I want to now gain a better ...
- 515
8
votes
0
answers
288
views
Very ample divisors on blow ups of the projective plane
Let $X$ be $\mathbb{P}^2$ blown up at $k$ points in general position. The Picard group of $X$ is just $\mathbb{Z}^{k+1}$ and one knows the intersection product explicitly. If $D$ is an ample divisor, ...
- 2,883
7
votes
0
answers
258
views
Cohomology of fibers of a morphism of a blowup of affine space
Consider $\mathbb A^n$ and let $\Sigma$ be a subdivision of its toric fan $\mathbb R^n_{\geq 0}$. This induces a toric blowup $\pi : Y \to \mathbb A^n$. Let $X \subseteq Y$ be the preimage of the ...
- 1,084
7
votes
0
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893
views
Is the Springer resolution a blow-up?
Let's consider the Springer resolution of the nilpotent cone $\mathcal{N}$ of a complex semisimple Lie algebra $\mathfrak{g}$, which is
$$
\widetilde{\mathcal{N}}=T^*\mathcal{B}\rightarrow \mathcal{N}...
- 8,707
6
votes
0
answers
342
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: \...
- 2,195
6
votes
0
answers
317
views
Does integral closure commute with pushforward
Suppose that $\pi : Y \to X$ is a proper birational morphism between normal varieties (schemes, whatever). Suppose that $I$ is an ideal sheaf on $Y$.
One can form $\pi_* I$ and construct an ideal ...
- 20.2k
6
votes
0
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289
views
blowups and group actions
Let $X$ be a smooth projective variety over the complex numbers and assume that $X$ is equipped with the action of a finite group $G$.
Denote by $Z$ the closed subscheme of fixed points of $G$ and ...
- 61
6
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0
answers
301
views
degenerating surface II
In degenerating surface, Robert Bryant give us an example of a sequence of minimal immersions which converges (in $C^2$- topology) to $z\mapsto z^{2k+1}$ on the unit disc $\mathbb{D}$. My question is ...
- 914
5
votes
0
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309
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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' \...
- 1,044
5
votes
0
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166
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 ...
- 1,338
5
votes
0
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279
views
When is the strict transform of very ample divisor ample?
Let $X$ be a projective variety and $Y \subset X$ is a very ample divisor on $X$. Let $Z \subset X$ be a regular subvariety (but $X$ need not be regular along $Z$) of codimension $2$ and $\pi:\tilde{X}...
- 2,126
5
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0
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210
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Nonlinear Schrödinger blow-up for non radial solutions
I am studying a paper of Frank Merle and Pierre Raphaël,
http://math.unice.fr/~praphael/Publications/blow-up-norme-critique.pdf.
The equations are
$$
i\partial_tu+\Delta u=-|u|^{p-1}u
$$
on $\mathbb{R}...
- 1,588
5
votes
0
answers
676
views
When is the proper transform of a subscheme equal to the total transform?
Say that $X \subset Y \subset \mathbb{P}^n$ are closed subschemes and let $\pi: \tilde{Y} \rightarrow Y$ be the blowup along $X$.
If $H$ is a hyperplane section of $Y$, under what conditions (on $H$...
- 213
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0
answers
3k
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cohomology of a blowup: reference needed
Does anybody know a reference in which the computation of the cohomology of a blow-up is made in detail?
- 51
4
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0
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236
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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 $...
- 243
4
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0
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195
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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 $C^{k, \alpha}$ ...
- 191
4
votes
0
answers
100
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. ...
- 1,111
4
votes
0
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126
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Diffeomorphism of $ \mathbb{C}P^2 \# ~\overline{\mathbb{C}P^2}$
I am currently reading Dusa McDuff's paper "Blow ups and symplectic embedding in dimension 4" and had a few questions regarding the paper.
In the paper McDuff uses the following notation. $X = \...
- 755
4
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answers
217
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Blow-up of $\mathbb{P}^4$ along a smooth surface
Let $\pi \colon X\to \mathbb{P}^4$ be the blow-up of a smooth surface $S\subset \mathbb{P}^4$. Is there a formula to compute $(K_X)^4$ ? (which should be dependent on invariants of $S$).
In dimension ...
- 7,652
3
votes
0
answers
151
views
Cotangent complex of a blowup
Let $X$ be a nonsingular variety over an algebraically closed field $k$, and let $Y \subset X$ be a nonsingular subvariety. Consider the blowup $p: \tilde{X} \to X$ of $X$ along $Y$, with exceptional ...
- 31
3
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179
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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{...
3
votes
0
answers
374
views
The Chow ring of a blow-up along a badly embedded subscheme
Let $X$ be a variety and $i:Y\hookrightarrow X$ be a regularly embedded closed subvariety, and write $\tilde X$ for $Bl_Y X$ the blow-up of $X$ along $Y$. Let $A^*(X)$ refer to the Fulton-Macpherson ...
- 518
3
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0
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141
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Which blow ups in the base of a conic bundle preserve the "standard" condition?
Assume we are given a nontrivial standard conic bundle $\pi: X\rightarrow S$, that is $X$ and $S$ are smooth projective varieties (say over $\mathbb{C}$), $\pi$ is flat and furthermore we have $Pic(X)=...
- 1,015
3
votes
0
answers
487
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the normalized blowup
Let $X$ be a normal variety over $\mathbb{C}$ and $x\in X$ a singular point.
Let $f:Y^{\nu}\to X$ be the normalized blowup at $x\in X$. (i.e. $f$ is a composition of the blowup $Y:=Bl_xX\to X$ and ...
- 31
3
votes
0
answers
808
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
0
answers
234
views
Jacobian Conjecture, Cubic-Keller maps
I have recently read an interesting article about the Jacobian Conjecture, in particular the reduction to the case $f(x) = x + A(x)^3$.
I was wondering about codimension one divisors on $Y = A^n$. ...
- 31
3
votes
0
answers
390
views
Blowing up a projective surface
Let $X$ be a smooth degree $d$ ($d>5$) surface in $\mathbb{P}^3$. We now blow up $X$ at a point, embed it in some projective space, and and consider a projection of it into $\mathbb{P}^3$. The ...
- 1,020
2
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0
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283
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Blow up at an ordinary double point
Let $X \subset \mathbb{C}^n$ be a complex complete intersection surface with only ordinary double point singularities. Let $o$ be such an ordinary double point.
Let $\tilde{X}$ be the strict transform ...
2
votes
1
answer
330
views
How to compute the transfer maps for G-theory of Noetherian schemes
Let $k$ be a field and $R$ be the ring $k[x,xy,xy^2,xy^3]$. Let $X$ be $\operatorname{Spec}(R)$ and $\tilde{X}$ be the blow-up of $X$ along the maximal ideal $I$ of $R$ generated by $x,xy,xy^2,xy^3$.I ...
- 531
2
votes
0
answers
195
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Is this blow-up a line bundle over the projective line
Let $R$ be the ring $\mathbb{C}[a,b,c,d]/(ac-b^2,bd-c^2,ad-bc)$. Let $I$ be the ideal of $R$ generated by $a,d$. Let $X=\operatorname{Spec}(R)$ and $\tilde{X}$ be the blow-up of the affine scheme $X$ ...
- 531
2
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0
answers
185
views
Blow-up at locally closed center
Let $X$ be a scheme, and $Z\subset X$ is a locally closed subscheme. I am wondering that is there a reasonable definition of blow-up of $X$ at $Z$?
And of course, this blow-up $Bl_Z X$ should satisfy ...
- 505
2
votes
0
answers
185
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 $2$. ...
- 2,022
2
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0
answers
413
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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, ...
- 2,022
2
votes
0
answers
183
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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 ...
- 393
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0
answers
284
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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
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0
answers
202
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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 ...
- 5,851
2
votes
0
answers
144
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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 ...
- 737
2
votes
0
answers
147
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Is this construction with stacks a blow-up?
Let $X$ be the stack of rank $1$ degree $b$ coherent sheaves $E$ with torsion of length at most 1 on an elliptic curve $C$. Let $Y$ be the stack of pairs $E^{'} \subset E$ such that $E \in X$ and $E/E^...
- 2,490
2
votes
0
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63
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Blowing up the base of an elliptically fibered (non Weierstrass) threefold
Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $\sigma_i$ ,$i=1 \dots n$. None of these "sections" are honestly a section, they ...
2
votes
0
answers
228
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Blow-up, strict transform and gysin map
Let $X$ be a non-singular, projective variety and $Y \subset X$ non-singular, projective subvariety. Denote by $\pi:W \to X$ the blow-up of $X$ along $Y$. Let $Y_1 \subset W$ be a non-singular ...
- 2,126
2
votes
0
answers
252
views
Blow-up along singularity of the degeneracy locus
Let $X$ be a smooth, projective variety, $E$ a rank $r$ locally free sheaf on $X$. Fix a closed embedding $i:X \hookrightarrow \mathbb{P}^N$ and denote by $\mathcal{O}_X(m)=i^*\mathcal{O}_{\mathbb{P}^...
- 1,949
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0
answers
117
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Quantitative Approach to Existence of Minimal-Mass Blowup Solutions to NLS
Consider the mass-critical defocusing NLS in dimension $d\geq 1$:
$$iu_{t}+\Delta u = |u|^{4/d}u, \quad (t,x) \in I\times\mathbb{R}^{2}$$
Define the mass $M(u)$ and scattering size $S(u)$ of the ...
- 863
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votes
0
answers
97
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Relation between symplectic blow-up of a compact manifold and fibre bundles over same manifold
The symplectic blow-up of a compact symplectic manifold $(X,ω)$ along a compact symplectically embedded submanifold $(M,σ)$ results in another compact manifold $(\tilde{X},\tilde{ω})$ given by
$$\...
- 191
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votes
0
answers
382
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Blow up along a section of a smooth morphism
Let $C$ be a ground locally notherian and quasiprojective scheme. Let $\pi:S\to B$ be a $C$-morphism of finite type $C$-schemes. We call $\pi$ a $C$-smooth morphism if the morphisms $S\to C$ and $B\to ...
- 43
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votes
0
answers
252
views
Is any blow-up of smooth subvarieties always an extremal contraction?
Let $X$ be a smooth complex projective variety and $Z$ be a smooth subvariety of $X$.
Take the blow-up $\pi: Y \to X$ of $X$ along $Z$.
Then I want to know whether $\pi$ is the contraction of an ...
- 31
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0
answers
521
views
higher direct images of O(E)
I hope this is well known, I just could not work it out myself.
Say I have a variety X (smooth and projective over C is my usual setup) with a smooth subvariety Z. Let f: BL_Z(X) --> X be the blowup ...
- 1,250
2
votes
0
answers
122
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Can discrepancy change after pseudo isomorphisms?
Let $X$ and $Y$ be two projective irreductive algebraic varieties of dimension $3$ and let $f:X\dashrightarrow Y$ be a pseudo isomorphism, i.e. a birational map which restricts to an isomorphism ...
- 7,652