Questions tagged [blow-ups]

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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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0 votes
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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 $Y$...
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  • 422
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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 \...
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  • 101
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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 \...
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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 $2$. ...
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  • 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 $...
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  • 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 $C\to\mathcal{X}$ ...
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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}_{\...
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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 ...
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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 ...
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  • 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(...
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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 ...
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  • 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 ...
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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 $C^{k, \alpha}$ ...
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2 votes
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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, ...
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  • 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 $\...
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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....
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  • 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 ...
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  • 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 \...
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  • 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 ...
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  • 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 ...
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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 ...
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  • 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 ...
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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 ...
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  • 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 $\...
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  • 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 ...
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  • 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 \...
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  • 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-...
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  • 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{...
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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 ...
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  • 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 ...
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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. ...
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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 ...
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  • 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' \...
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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 ...
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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 ...
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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 $F$...
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  • 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 ...
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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 ...
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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 ...
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  • 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 ...
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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\...
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  • 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: \...
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  • 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 ...
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  • 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 $T'=\{q_1,\ldots,q_5\}$ ($T'\neq T$), ...
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  • 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 $Bl_Y X$. ...
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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 ...
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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 ...
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  • 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 ...
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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 ...
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