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Questions tagged [blowup]

A technique in geometry (especially algebraic and differential, and, by extension, the study of pseudo-differential operators) for resolution of singularities. Not to be confused with the formation of singularities in solutions of ordinary or partial differential equations.

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What is the meaning of $\cup _{\pi}$

In Griffith-Harris' definition for the blow-up (p. 182 in my version) they define the blow-up at some point $x$ with the restricted projection map $\pi : \widetilde{\Delta} \setminus E \rightarrow U \...
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Blow Up of a Surface

I have a question a step in the example demonstating the blowing up in Liu's "Algebraic Geometry and Arithmetic Curves" in the excerpt below (or look up at page 320): We blow up the surface $X= ...
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Blow up at point is finite?

Let $X$ be an affine algebraic curve with $0 \in X$ and $\tilde{X}$ the strict transform of $X$ w.r.t the blowup of $X$ at $0$. How to prove that $\pi \colon \tilde{X} \to X$ is finite? Is it even ...
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Blowup of non-singular varieties

How to construct a blowup $f$: X $\rightarrow$ Y between non-singular quasi-projective varieties? This is an exercise from Shafarevich "Basic Algebraic Geometry 1". I want to construct for any $n \...
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Difference between blowup of $V_1\cup V_2$ and union of blowup of $V_1$ and blowup of $V_2$

Let $V_1,V_2\subset \mathbb C^N$ be sub varieties and let $C=\{f_1=\dots=f_d=0\}$ be a subvariety of $\mathbb C^N$. Consider the blowup of $\mathbb C^N$ along $C$ and denote it by $B_C\mathbb C^N$. ...
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Chern class of tautological line bundle over the projectivization of a vector bundle

Let $\mathbb{C}^k\hookrightarrow E\to B$ be a complex vector bundle. Let $\mathbb{CP}^{k-1}\hookrightarrow\mathbb{P}(E)\to B$ be its projectivization. We can consider the tautological line bundle $L$...
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Counter example on continuity of blow-up time with respect to parameters

Consider the following differential equation (DE) $$\dot{x}(t)=ax(t)+f(x(t))+u(t).$$ Let assume there is number $M$ such that the DE admits a local solution over some interval $[0,\tau]$ for all $|u(t)...
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Why is this construction a complex manifold?

I am beginning to study blow-ups, and in the development of the blow-up of $\mathbb C ^2$ in the origin, the author claims without further clarification that the following construction yields a ...
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Universal property of blow up of complex analytic space

I know that there is a universal property of blow-ups in the algebraic setting (see Wikipedia). How does this translate to the case of complex geometry and holomorphic/bimeromorphic maps? I am ...
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rational map on curves coincide iff tangent

The following is an exercise from Shafarevich. Let $\varphi: \mathbb{P}^2\rightarrow\mathbb{P}^4$ be the rational map defined by $$\varphi(x_0:x_1:x_2) = (x_1x_2:x_0x_2:x_0x_1).$$ Consider the ...
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Blow up of plane curve is Normalization of local ring?

I have a question concerning normalizations of plane curves, which I know little about. Consider the simple node $V(f = y^2 - x^3 - x^2)$. Then $t=y/x$ is integral over $k[x,y]$ so that $(k[x,y]/f) \...
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Blow-up of affine space along subvariety

Brief summary of the question: Let $C\subset \mathbb A^n$ be a singular curve and $\pi:X=Bl_C\mathbb A^n\to \mathbb A^n$ be the blow-up along $C$. 1) Is there a reference showing that $\pi^{-1}...
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Canonical bundle of blow up at singular point

Let $X$ be a complex variety/ manifold with one singular point $x_0\in X$. If we blow up $X$ at $x_0$, we obtain a smoot variety/manifold with exceptional divisor $Y$. How can we calculate the ...
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82 views

Holomorphic form an a blow up

Consider the blow up of $\mathbb C^2/\mathbb Z_2$ at its singularity $0$. Since $dz_1\wedge dz_2$ is invariant under $z\mapsto -z$, it passes to a well defined holomorphic form on $(\mathbb C^2/\...
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Intersection of the strict transform of a curve and exceptional hypersurface in the blow up of the affine plane.

Consider the curve $Y$ to be $(\frac{t^3}{1-t},\frac{t^4}{1-t})$ with $t$ different from 1. I need to parametrize the curve, calling $x$ the first coordinate ad $y$ the second we find that $t^3=x(1-t)$...
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Resolution of singularity of $\mathbb C^2/{\mathbb Z_2}$ (blow up)

Consider the $\mathbb Z_2$-action $g:\mathbb C^2\to \mathbb C^2, z\mapsto-z$ on $\mathbb C^2$ and its quotient $X:=\mathbb C^2/{\mathbb Z_2}$. This is a singular surface with singular point the image ...
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The valuation attached to a smooth point of an algebraic variety

I am interested in proving the following result: Let $k$ be an algebraically closed field, $X$ a normal integral variety over $k$ and $x\in X$ a closed point. Write $\mathfrak{m}_x$ for the ...
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Blow-up conditions for an ODE with perspective nonlinearity

I would like to obtain necessary and sufficient conditions on $f$ (within a certain class of $f$, defined below) for finite-time blow-up of ALL positive solutions of the non-autonomous ODE $$x^{\...
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78 views

When is the blow up morphism flat?

The question is as in the title: given a scheme $X$ and a closed subscheme $Z$ when is it true that the blow up morphism $Bl_ZX \rightarrow X$ is flat? I’m mainly concerned with $X$ being a smooth ...
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Understanding the blow up of $\mathbb{A}^2$ in $\left<x_0,x_1\right>$.

The following is an example from Gathmann's notes on algebraic geometry: I am having problems with showing, rigorously, that $\tilde X$ is given by the prescribed equations. First, I do not get why ...
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The pullback line bundle restricted on the exceptional divisor is trivial

Let $\sigma:\hat X\to X$ be the blow up of a point $x\in X$, denote the exceptional divisors $\sigma^{-1}(x)$ by $E$. $L\to X$ is a line bundle. Then we have a pullback line bundle $\sigma^*L\to\hat X$...
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Calculation of valuation ring of a valuation associated to a blowup

Let $\mathfrak{m} = (x,y) \subset k[x,y]$. Then the valuation $v$ of $k(x,y)$ associated to the exceptional divisor of the blowup should be defined by $$v(f) = \mathrm{sup}(n|f \in \mathfrak{m}^n), f\...
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Section of the Sheaf of Holomorphic Functions

I'm reading now the book "Vector Bundles on Complex Projective Spaces" from Okonek, Schneider and Spindler and I have an understanding problem with the interpretation of a section in following excerpt ...
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blowing-up preserves the first Betti number?

Let $G$ be a finite group and $S$ be a K3 surface. $G$ acts effectively and symplectically(fix the nowhere vanishing 2-form of $S$) on $S$. Since the action is symplectic,quotient surface $S/G$ has ...
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Blow-up and gluing coordinates

I am reading the book "Algebraic Geometry and Statistical Learning Theory" by Sumio Watanabe and have a question regarding Remark 3.16 (1) on page 95. He defines the blow-up of $ \mathbb{R}^2$ with ...
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Genus of the desingularization of a plane curve through meromorphic function

Could someone help me? Consider, in the complex projective plane, the curve $C$ given by the the points $[X,Y,Z]$ for which $F(X,Y,Z)=X^2Y^3+YZ^4+Z^5=0$. I have to desingularize the curve and consider ...
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82 views

Definition of blowing-up along a complex submanifold (Huybrechts)

I am trying to understand Huybrecht's definition of the blow-up of a complex manifold $X$ along a submanifold $Y$ - if you don't have a hard copy to hand, I have found an electronic version here (see ...
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How to show $CP^1\times CP^1$ and $CP^2$ blow up one point are not diffeomorphic?

How can we prove $CP^1\times CP^1$ and $CP^2$ blow up one point is not diffeomorphic? I tried to compute their Hodge numbers and Chern numbers but they are the same.
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Blow up of one point is isomorphic to $\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(1))$

My question comes from an example in Hartshorne (Example V.2.11.4) which I'm having trouble following. It is claimed that the Blow up of a point $p \in \mathbb{P}^n$ is isomorphic to $\mathbb{P}(\...
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Euler Characteristic of a blowing up

I'm trying to compute the Euler Characteristic of the blowing up of $\mathbb{C}\mathbb{P}^2$ at $n$ points. Does anyone know how could I do this?
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186 views

Restriction of the exceptional divisor to itself as a line bundle

If we take a complex projective variety $X$ and blow it up at a point, we get an exceptional divisor $E\cong \mathbb{P}^{n-1}$, where $n=dim(X)$. My question basically regards $\mathcal{O}_{\tilde X}...
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Blow up of $\mathbb{P}^2$ in a point and direct image sheaves

I am trying to understand better direct image sheaves. To do so, I want to start working in a particular and easy example. Let $\pi:X\rightarrow \mathbb{P}^2$ be the blow up of $\mathbb{P}^2$ in a ...
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Expression for $\mathcal{O}_{\mathbb{P}(N_{Y/X})}(-k)$ in blowup

Let $X$ be a smooth toric variety and $Y \subset X$ be a complete intersection with the normal bundle $N_{Y/X}$ and $E \subset \mathrm{Bl_Y X}$ be exceptional divisor. Than it is well known that $E \...
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Is it possible to blow-up in codimension one?

In the context of complex manifolds, one can consider a blow-up along a complex submanifold. For a linear subspace of $\mathbb{C}^n$ there is a general procedure to perform such a blow-up: for a ...
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Example of blow-up of curve along subvariety with finite fields

I would like to work through computing the blow-up of a particular curve along a subvariety consisting of just two points. Let $$F(X,Y,Z) = X^4 + Y^4 - X Y^2 Z - X^2 Z^2 + Y^2 Z^2 + Z^4 \in \mathbb{F}...
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$K_0$ ring of del-pezzo surface

Is there book or paper where described structure of $K_0$ ring of a del-pezzo surfaces? Especially in case of the blowup of projective line in three points. I know method for computing $K_0$ ring of ...
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Blow-up of the affine plane in the origin, using schemes.

I am working through section IV.2 about blow-ups of the book Geometry of schemes by Eisenbud and Harris. I am having some trouble understanding the details of the following example, which they discuss ...
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How can I get claim 1 or claim 2?

I have such an equation; When I mutliply -2Δut, I get energy inequality. \begin{align} & u_{tt}- \Delta u + a \Delta^{2} u = b f (-\Delta u), \quad (t,x) \in (0,T) \times (\Omega ∪ \partial \...
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81 views

Ideal sheaf of intersection

Let $p_1,p_2,p_3$ be the points in the general position in $\mathbb{P}^2$ and $\mathcal I$ be their ideal sheaf. I want to find locally free resolution of $\mathcal I$. I can write $0\to\mathcal{O}(-2)...
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Affine cover of blow-up along ideal

I would like to find affine cover of blow-up $X = Bl_{I} \mathbb A^2$, where $I=(x^3, xy, y^2)$. I know that $X=\{((x_1, x_2),[y_1,y_2,y_3])\subset \mathbb A^2\times\mathbb P^2: x_1^3y_2=x_1x_2y_1, ...
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81 views

Images of derived categories of $X, Z$ in derived category of blow up

Let $X$ be a smooth variety and $Z \subset X$ be a locally complete intersection (smooth if needed). So $X, Z$ is as good as we need (i am working with toric varieties). Let $\pi : \mathrm{Bl}_Z X \to ...
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Cohomology of blow-up

I read from somewhere the following proposition (maybe more condition needed): Let $X$ be a $\mathbb C$-scheme, $Z$ its closed subscheme. Let $B=Bl_Z X$ be the blow-up along $Z$ and $E$ be the ...
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Is $\Bbb P^1_k$ simply the blow up of $\operatorname{Spec}k$?

I was playing around with the definition of a blow up when I encountered something interesting. Theorem IV-23 Eisenbud & Harris Let $X$ be a scheme and $Y\subset X$ a closed subscheme. Let ...
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Prove that $ C_1'\cap Z=C_2'\cap Z $ if and only if $ C_1 $ and $ C_2 $ touch at $ \xi $.

This is Exercise II.4.1 in Shafarevich's book Basic Algebraic Geometry, second edition. Suppose that dim $ X = 2 $ and that $ \xi \in X $ is a nonsingular point. Let $ C_1, C_2 \in X $ be two ...
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About blow-up of segre variety

As an example, I want to find resolution of singularity of Segre variety in $M_{2,2}$, let this variety $V=V(x_1x_4-x_2x_3)$. Then blow-up of $V$ at $O$ is a closed subset of $\mathbb{A}^4 \times \...
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94 views

Transformation of differential operators by blowing up

M is a smooth space, $\sigma:M'\rightarrow M$ the blowing-up, centre is a smooth closed subspace C. A local coordinate chart U of M with coordinates $(x_1,...,x_n)$ in which $C=\{x_r=...=x_n=0\}$. ...
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1answer
156 views

Self-intersection number of the proper transform of the line at infinity

Reference and Question 7(a) of Problem Set 8 The main parts of Question 7(a) are: Let $H$ be the line at infinity in $\mathbb{CP}^{2}$, and let $P$ and $Q$ be distinct points on $H$. Let $X$ be the ...
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158 views

Is the dual of the normal bundle of the exceptional divisor ample

Let $X$ be a non-singular projective variety and $Y \subset X$ a non-singular subvariety. Denote by $\tilde{X}$ the blow-up of $X$ along $Y$ and $E$ the associated normal bundle. Denote by $N_{E|\...
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491 views

Is there a relation between the cohomology ring of a blowup with the base scheme and blowup locus?

Let $$ Y \subset X $$ be a codimension 2 or greater smooth subvariety of a smooth projective variety $X$. Is there a relation between the cohomology of the blowup $Bl_Y(X)$ and the cohomology rings of ...
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110 views

Exceptional divisor as projectivization of the tangent space

Let $X\subseteq \mathbb{P}^n$ be a projective variety, and let $p\in X$. I define the blow up of $X$ at $p$ as the closure $\Gamma$ in $X\times \mathbb{P}^{n-1}$ of the graph of the projection $\phi$ ...