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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 ...
Hephaistos's user avatar
0 votes
0 answers
64 views

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 ...
Hephaistos's user avatar
2 votes
0 answers
169 views

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 ...
Serge the Toaster's user avatar
7 votes
0 answers
223 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 ...
Leo Herr's user avatar
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0 answers
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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 ...
Boris's user avatar
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1 vote
0 answers
42 views

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:=\...
Display Name's user avatar
2 votes
1 answer
258 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 ...
Boris's user avatar
  • 331
0 votes
1 answer
181 views

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 ...
user267839's user avatar
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1 vote
1 answer
172 views

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 ...
user267839's user avatar
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2 votes
0 answers
163 views

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 $X$ ...
Boris's user avatar
  • 331
2 votes
2 answers
339 views

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 ...
Hydrogen's user avatar
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1 vote
0 answers
132 views

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 $X$?
Andrey Ryabichev's user avatar
1 vote
0 answers
98 views

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}...
Boris's user avatar
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2 votes
0 answers
169 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 ...
Kim's user avatar
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1 vote
0 answers
127 views

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 ...
Yachen Liu's user avatar
2 votes
1 answer
146 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 ...
user302934's user avatar
0 votes
0 answers
178 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 $Y$...
KingVon's user avatar
  • 428
0 votes
1 answer
139 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 \...
Serge the Toaster's user avatar
2 votes
0 answers
139 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 $2$. ...
Jana's user avatar
  • 2,002
4 votes
0 answers
217 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 $...
calc's user avatar
  • 243
1 vote
0 answers
95 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 $C\to\mathcal{X}$ ...
Nanjun Yang's user avatar
6 votes
1 answer
401 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}_{\...
Ben C's user avatar
  • 2,549
0 votes
0 answers
116 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 ...
gigi's user avatar
  • 1,303
2 votes
1 answer
208 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 ...
gigi's user avatar
  • 1,303
2 votes
1 answer
142 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(...
Bobech's user avatar
  • 371
9 votes
1 answer
703 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 ...
Student's user avatar
  • 4,570
9 votes
0 answers
298 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 ...
Andy's user avatar
  • 515
4 votes
0 answers
156 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 $C^{k, \alpha}$ ...
user734979's user avatar
2 votes
0 answers
273 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, ...
Jana's user avatar
  • 2,002
2 votes
1 answer
208 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 $\...
user avatar
5 votes
1 answer
243 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....
Ron's user avatar
  • 2,096
2 votes
0 answers
217 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 ...
user69559's user avatar
  • 1,483
11 votes
1 answer
1k 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 \...
Lineer 's user avatar
  • 388
2 votes
0 answers
165 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 ...
Li Li's user avatar
  • 355
2 votes
0 answers
207 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 ...
Blow's user avatar
  • 21
2 votes
1 answer
176 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 ...
Leo Herr's user avatar
  • 1,004
2 votes
0 answers
188 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 ...
user127776's user avatar
  • 5,607
1 vote
0 answers
159 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 ...
user127776's user avatar
  • 5,607
2 votes
1 answer
225 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 $\...
Jana's user avatar
  • 2,002
2 votes
1 answer
248 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 ...
gigi's user avatar
  • 1,303
9 votes
5 answers
1k 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 \...
user20107's user avatar
  • 177
7 votes
1 answer
390 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-...
gigi's user avatar
  • 1,303
3 votes
0 answers
165 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{...
Federico Fallucca's user avatar
7 votes
1 answer
398 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 ...
Jana's user avatar
  • 2,002
4 votes
1 answer
358 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 ...
Mingchen Xia's user avatar
4 votes
0 answers
95 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. ...
Carlos Esparza's user avatar
5 votes
1 answer
209 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 ...
Leo Herr's user avatar
  • 1,004
5 votes
0 answers
264 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' \...
Roberto Pignatelli's user avatar
2 votes
0 answers
126 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 ...
Hajime_Saito's user avatar
1 vote
0 answers
111 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 ...
Walter Simon's user avatar