# Questions tagged [blow-ups]

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199
questions

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### How to compute direct images for a blowing up?

Let $X$ be a smooth algebraic variety, $Z\subset X$ a smooth closed subvariety, and
$\pi:\tilde{X}\to X$ the blowing up of $X$ along $Z$. Let $E\subset\tilde{X}$ be the exceptional divisor, and $n$ an ...

0
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### Direct image of sheaves for blowing ups [duplicate]

Let $X$ be a smooth algebraic variety, $Z\subset X$ a smooth closed subvariety, and
$\pi:\tilde{X}\to X$ the blowing up of $X$ along $Z$. Let $E\subset\tilde{X}$ be the exceptional divisor, and $n$ an ...

2
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0
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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 ...

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### 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 ...

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### How to show the inclusion of the exceptional divisor is the zero section of the line bundle

Let $k$ be a field and $R$ be the ring $k[x,xy,xy^2,xy^3]$. Let $I$ be the ideal of $R$ generated by $x,xy,xy^2,xy^3$.Let $X$=Spec$(R)$ and $\tilde{X}$ be the blow-up of $X$ along $I$.I managed to ...

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### Points in a blowing up determined by an infinitesimal curve not in the centre

Let $\Bbbk$ be a field. Let $A$ be a $\Bbbk$-algebra (commutative and unital). All ring maps are $\Bbbk$-algebra homomorphisms. Let $I\subseteq A$ be an ideal. Let $X:=\mathrm{Spec}(A)$ and $Z:=\...

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258
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### 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 ...

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181
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### Calculate blowup of a pencil of cubics "by hand"

I have one more question about the Example (I.5.1) on page 7 from
Rick Miranda's the basic theory of elliptic surfaces:
Let $C_1$ be a smooth cubic curve in $\mathbb{P^2}$ and let $C_2$
be any other ...

1
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1
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172
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### Blow-up of a pencil of cubic curves (from Miranda's basic theory of elliptic surfaces)

Rick Miranda's "The basic theory of elliptic surfaces" the Example (I.5.1) see page 7 on a pencil of plane curves contains an argument that I do not understand yet.
Let $C_1$ be a smooth ...

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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 $X$ ...

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2
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339
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### Blowing up of a singular subvariety

I ask the same question on MathStackExchange but receive no answer.
I'm reading Kollár Mori Chapter 2.3. And they state the following lemma:
Lemma 2.29. Let $X$ be a smooth variety and $\Delta = \sum ...

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0
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132
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### Do blowups generate the birational equivalence relation?

Suppose $X$ and $Y$ are birational varieties (projective, irreducible, over $\mathbb C$) of the same dimention. Is there a sequence of blow-ups and blow-downs that makes $Y$ from $X$?

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### How to compute the G-theory groups of a blow-up of Noetherian schemes

Suppose that $k$ is an algebraically closed field and $R$ is a finitely generated $k$-algebra such that if $X$ denotes Spec$R$, then the only closed, singular point of $X$ is the origin. Let $\tilde{X}...

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### 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 ...

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### Blow up singularities on curves

Let $p$ be a prime number and let $\bar{\mathbb F}_p$ be an algebraic closure of $\mathbb F_p$. Let $C$ be an irreducible singular projective curve over $\bar{\mathbb F}_p$.
Let $P$ be a singularity ...

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### 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 ...

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178
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### 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 $Y$...

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139
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### 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
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### 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 $2$. ...

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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 $...

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### 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 $C\to\mathcal{X}$ ...

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### 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}_{\...

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### 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 ...

2
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1
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208
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### 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 ...

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### 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(...

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### 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 ...

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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 ...

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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 $C^{k, \alpha}$ ...

2
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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
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1
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### 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 $\...

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### 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....

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### 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 ...

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### 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 \...

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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 ...

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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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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 $\...

2
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1
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### 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 ...

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### 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 \...

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### 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-...

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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{...

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### 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 ...

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1
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### 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 ...

4
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0
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### 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. ...

5
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1
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### 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 ...

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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' \...

2
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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 ...

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### 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 ...