# Questions tagged [blow-ups]

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154
questions

**4**

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129 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**

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87 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 ...

**1**

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62 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 ...

**2**

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**1**answer

164 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 $F$...

**6**

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**2**answers

184 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 ...

**2**

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66 views

### Birational contraction of toric vector bundle

Let $X$ be the toric vector bundle over $\mathbb{P}(1,1,1,2)$ with grading matrix
$$
\left(\begin{array}{cccccc}
1 & 1 & 1 & 2 & -2 & 0 \\
0 & 0 & 0 & 0 & 1 & ...

**5**

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108 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 $...

**1**

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**1**answer

92 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.
...

**3**

votes

**1**answer

201 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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130 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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173 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: \...

**10**

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157 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 ...

**8**

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**2**answers

392 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 $T'=\{q_1,\ldots,q_5\}$ ($T'\neq T$), ...

**0**

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92 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 $Bl_Y X$. ...

**6**

votes

**1**answer

407 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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63 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 ...

**7**

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**1**answer

262 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 ...

**1**

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**1**answer

126 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 ...

**4**

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115 views

### 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 = \...

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123 views

### 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^...

**1**

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**1**answer

211 views

### Blow up of a Projective scheme along a projective-subscheme using Macaulay2

Given a closed-subscheme (corresponding to the homogeneous ideal $I$ inside $K[x_0,...,x_n]$) of a projective scheme $\mathbb{P}^n$, how can one find out the blow up using Macaulay2? ...

**2**

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**1**answer

186 views

### Counter example about blow-up solution of DEs

Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...

**8**

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164 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**

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60 views

### 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 ...

**3**

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**1**answer

147 views

### Gradient blowup for the 1-dim heat equation near an irregular boundary point

I'm trying to estimate the rate of boundary gradient blow-up for the 1-dim heat equation near an irregular boundary point.
Let $b(t) := (1-t)^\alpha$, $\alpha < 1/2$. Let $u(t,x)$ solve the ...

**3**

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218 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 ...

**2**

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**1**answer

449 views

### An explicit computation of the blow-up of curve over $\mathbb{F}_3$ at two points

I would like to work through computing the blow-up of a particular curve along a subvariety consisting of just two points, both of which are ordinary double points.
Let $$F(X,Y,Z) = X^4 + Y^4 - X Y^2 ...

**3**

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689 views

### Topology of the blowup of a surface at a point (connected sum)

Let $S$ be a complex algebraic (smooth) surface and $\widetilde{S}$ be the blowup of $S$ at a point $p\in S$.
I would like to understand the statement:
As a topological manifold, $\widetilde{S}$ ...

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45 views

### How can I describe in explicit geometric terms the (in general non-complete) linear system?

Let $\varphi\!: S \to S^\prime$ be a birational morphism of projective non-singular irreducible surfaces over a (algebraically closed) field $k$ and let $D \in \mathrm{Div}(S)$. Also, let $\big(\...

**3**

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189 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}...

**1**

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**1**answer

117 views

### Gysin map for projective sub-bundles of exceptional divisors

Let $X$ be a smooth, projective variety, $Y \subset X$ a smooth, projective subvariety of codimension $3$. Denote by $\pi:\tilde{X} \to X$ the blow-up of $X$ along $Y$ and by $E$ the exceptional ...

**1**

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451 views

### Cotangent sheaf of a blow up

It's well known that for a nonsingular variety $X$ and a nonsingular subvariety $Y$ with $\text{codim}(Y,X) \ge 2$, we can compute the canonical sheaf of $\widetilde{X}$ (the blow up of $X$ along $Y$) ...

**6**

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**1**answer

156 views

### Blowing-up an ideal generated by squares

Let $f_1,\dots,f_r$ be regular functions on a smooth projective variety $X$, and consider the ideals $I = (f_1^2,\dots,f_r^2)$ and $J = (f_1,\dots,f_r)$. Let $Y = Z(I)$ and $W = Z(J)$ be the ...

**13**

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391 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 ...

**2**

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141 views

### 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 ...

**4**

votes

**2**answers

285 views

### Is this divisorial contraction a blow-up?

Let $C$ be a curve in a smooth $3$-fold $X$ with an ordinary node $p\in X$. Blow-up $p$ let $E$ be the exceptional divisor, and $\widetilde{C}$ the strict transform of $C$. Furthermore let $L$ be the ...

**4**

votes

**1**answer

245 views

### Gysin map and blow up

Let $X$ be a smooth projective variety and $W \subset X$ a smooth, projective subvariety. Let $\pi:\tilde{X} \to X$ be the blow-up of $X$ along $W$. Let $E$ be the exceptional divisor of $\pi$ and $i:...

**10**

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**1**answer

754 views

### Blow-ups, pullbacks and proper transforms

Let $X$ be a smooth projective variety, $Z$ a smooth subvariety of $X$, and let $f:\widetilde{X}\to X$ be the blow-up of $X$ along $Z$. Then for a subvariety $V\subset X$, we have two cohomology ...

**0**

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**1**answer

98 views

### Critical gKdV - tutorial

I'm looking for a good introduction to the critical generelized KdV equation
$$u_t +u_{xxx}+5u^4u_x = 0 \, , $$
$$ u(t=0,x) = u_0 (x) \, , \qquad x\in \mathbb{R} \, ,$$
and its blowup solutions. There ...

**2**

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182 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}^...

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101 views

### 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 ...

**7**

votes

**1**answer

528 views

### Resolution of an isolated cyclic quotient singularity

I am looking for a reference to the following fact which seems to be true and which is probably well-known (at least to experts in resolution of singularities):
Consider an isolated cyclic quotient ...

**2**

votes

**1**answer

181 views

### blow up in finite time of hyperbolic system of conservation law

Let say I have a hyperbolic system of conservation law. How do I show there is a blow up in finite time? For a single conservation law, I think, I could just show that there is collision of ...

**1**

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151 views

### On simultaneous resolution of singularities in certain flat families

Let $X$ be a smooth projective variety (over the complex numbers) of dimension at least $2$, $B$ a finite set of closed points. Consider the closed subscheme $E:=B \times X + \Delta \subset X \times X$...

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115 views

### 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)=...

**8**

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**1**answer

2k views

### Blow-up in family

Let $\pi \colon X \to T$ be a flat projective morphism, and let $Y$ be a closed sub-scheme of $X$ which is flat over $T$. We can assume that everything is defined over the complex numbers, and $T$ is ...

**1**

vote

**1**answer

71 views

### Analytic conjugacy of vanishing holonomy groups implies analytic conjugacy of foliations

i am reading and trying to do some exercises and problems of the book Lectures on Analytic Differential Equations- Y. Ilyashenko, S. Yakovenko.
I can not solve the problem 11.6 that says
Consider ...

**4**

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**1**answer

170 views

### Does $u_{t}=g(t)u_{x}^{2}$ blow-up for bounded positive g? What about $u_{t}=u_{xx}+g(t)u_{x}^{2}$?

My original problem is to see if the following pde develops blow-ups in $(-L,L)$
$$u_{t}=u_{xx}+g(t)(u_{x})^{2}$$
for periodic boundary $u_{0}(-L)=u_{0}(L)$, where $0<g(t)<1$; specifically $g(...

**5**

votes

**1**answer

351 views

### What is the relation between holomorphic blow-up and symplectic blow-up?

McDuff has shown us exactly how the symplectic blow-up procedure along a symplectic submanifold affects the symplectic structure in the ambient space, i.e., if $\omega$ is the original symplectic form,...

**6**

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**1**answer

869 views

### Resolution of the $E_8$ singularity with a weighted blowup

I am reading Miles Reid's notes on weighted projective spaces, and I'm a little confused about a particular paragraph (notes here, page 8):
A famous case is the $E_8$ singularity $X: (x^2+y^3+z^5=0)...