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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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Is every normalization a blowup?

Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example. Let $Y \to X$ be the normalization. The answer is positive in ...
SeparatedScheme's user avatar
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Cones of divisors on $\text{CDiv}(X)_{\mathbb{R}}$

Let $X$ be a normal, projective $\mathbb{Q}$-factorial variety over the complex numbers. I know that, in order to understand the birational geometry of $X$, one may consider several cones of divisors, ...
ark's user avatar
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$A_1$ Du Val singularity and blow ups

Blowing up $\operatorname{Spec}K[x,y]=\mathbb{A}^2$ along $\operatorname{Spec}K[x,y]/(x^2,y)$ in the $U_B$ chart gives a singular point $U_B=\text{Spec}K[x,y][a]/(ya-x^2)$ at the origin. This ...
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Blowup of diagonal in $\mathbb{P}^r \times \mathbb{P}^r$

Let $X= \mathbb{P}^r \times \mathbb{P}^r$. Suppose we blowup $X$ along $\Delta$ the diagonal to get $\tilde X$. I want to show that this is isomorphic to the fibre product which I describe below - Let ...
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Connection between Chern classes and Blow ups

I know that a very important topological invariant in Complex Geometry and Algebraic Geometry is the Chern class of a vector bundle. Recently, I came across a paper discussing a potential connection ...
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Rees Algebra Isomorphism $A[xt, yt] \cong A[X, Y] / (yX - xY)$

David Eisenbud's book "The Geometry of Schemes" states the following: Proposition IV-25: Let $A$ be a Noetherian ring and $x, y \in A$; let $B$ be the Rees algebra $$ B = A[xt, yt] \subset A[...
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Blowing Up along Reduced vs Non-Reduced Subschemes

In Eisenbud's book, 'The Geometry of Schemes' (see Proposition IV-40), he demonstrates a connection between blowing up schemes along reduced and non-reduced subschemes. Specifically, he illustrates ...
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Why are local models of blow-ups compatible by gluing with transition functions in the normal bundle?

I am trying to understand the blow-up along a submanifold as explained in Huybrechts "Complex Geometry An Introduction", p. 99, Example 2.5.2. For background, for $m \leq n$, we see $\mathbb ...
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Blow up of Limao $x^2 - y^3z^3$ in $\mathbb{A}^3$

Let me take $X:=x^2-z^3y^3=0$ to be our surface over $\mathbb{A}^3$. Then we see we have a singular locus on the y and z axis. So, I will blow up initial on the Z-axis $V = Z(<x,y>)$. Then $\...
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Topology of blow up of a cone

Let $X$ be a real algebraic surface with a unique $A_1$ point at $p \in X$ (that is, the germ of $X$ at $p$ is isomorphic to the germ of $\{x^2+y^2=z^2\}$ at the origin). Let $Y \to X$ be the blow up ...
Serge the Toaster's user avatar
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Self-intersection of exceptional divisor of blowing-up along a singular point

Let $X$ be an $n$-dimensional projective variety with a simple singularity $p \in X$ which can be resolved by a blow-up $\pi\colon \tilde X \to X$ along $p$. The examples that I am considering are ...
Skadiologist's user avatar
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Exercise on blow up and strict trasformation

I am studying algebraic geometry from the Gathmann’s notes and I have to resolve exercise (it is not on the notes) Resolve the singularities of the following curve in $\mathbb{A}^2$ by subsequent ...
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Simple normal crossings divisor and blow up

I have to prove some things about simple normal crossings divisors and the blow up. The definition of a simple normal crossings divisor we use is a finite union $V = \cup_i V_i$ of irreducible quasi-...
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Transversality of strict transforms

When reading a book, I encountered the following assertion: let $k$ be a field (maybe perfect if it makes things easier), $X$ a smooth $k$-variety and $Y$ a closed irreducible subvariety, also smooth. ...
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Blow-ups and special fibers of schemes over DVR

Let $S$ be the spectrum of a DVR with generic point $\eta$ and closed point $s$. Let $X$ be a flat, quasi-projective scheme over $S$. Let $X_s$ denote the special fiber, and let $Z \subset X_s$ be a ...
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Definition and computation of concrete blow-up on projective plane

I have trouble finding a definition for the blow of of a point on the projective plane (or any projective space), and a projective plane curve. So first of all if you have a reference treating this I ...
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Example of blowup calculation given the definition

I am studying this article, but I did not understood the example of the blow up considering the definition of blow up given. In this part of the article, there is a definition: Consider a real ...
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Blow up of $\Bbb{A}^2$ at $(x^2,xy,y^2)$ is the same as at $(x,y)$

\begin{align*} \widetilde{\Bbb{A}^2}\subset\{(x,y)\times[y_1:y_2:y_3]\in\Bbb{A}^2\times\Bbb{P}^2:x^2y_2=xyy_1,xyy_3=y^2y_2,y^2y_1=x^2y_3\}=: Y. \end{align*} Consider the affine subset $U_i=\{(x,y)\...
Gabrielle Rodriguez's user avatar
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Blow up and hyperelliptic curves.

Let $C'$ be a non singular affine curve $y^2=x^5+3$ over $\mathbb{C}$. $C'^\#$ be its projective closure : $Y^2Z^3=X^5+3Z^5$. It has singular point at $\mathcal{O}'=(0:1:0)$. On the other hand, let $C$...
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Definition of the generic rate of a blow-up

I am reading a paper from van den Berg, Hulsof and King about blow-up solution for the harmonic map heat flow onto the sphere in a radially symmetric domain, this equation is given by, $$\theta_t = \...
Falcon's user avatar
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bigness of divisors under blow-up of finitely many points of the plane

Let $S\subset\mathbb{P}^2$ be a finite set of points in the plane, consider the blow-up of $\mathbb{P}^2$ along $S$: $$X=Bl_S(\mathbb{P^2})\rightarrow\mathbb{P}^2$$ and denote the divisor $D=2H-E$, ...
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Study of the properties of a non-local ODE

I am studying the following non-local ODE $$\dot p(x) \nu_{\varepsilon, \alpha}(x) + \int_{x}^{2x_0}\frac{\dot p(s)}{s + \varepsilon} ds = c \quad \text{for } x \in [0,2x_0].$$ The number $x_0$ can ...
Falcon's user avatar
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Confusion about line bundles and the intersection product

Let $X = \mathbb{A}_{\mathbb{C}}^{2}$ and let $Y$ be the blowup of $X$ at the origin. Let $E \cong \mathbb{P}^{1}$ the exceptional divisor. I think that we have a canonical inclusion $\mathcal{O}_{Y} \...
Fraktale Fatalität's user avatar
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Liu, Proposition 8.1.12: inverse image of blown-up ideal sheaf is invertible

I am struggling with Proposition 8.1.12e of Liu's Algebraic geometry and arithmetic curves. The setup is as follows: let $X=\text{Spec}A$, $I=(f_1,\dots,f_n)\subseteq A$ an ideal, and let $$\tilde A=\...
woolly-minded's user avatar
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Blowing up Nodal Curve

Following the first definition on page 3 of 1, the blow-up is defined as a closure of the image set in the corresponding product space. I have seen a few examples of this, for instance, the blow-up of ...
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Factorising a multivariate polynomial in terms of products of linear polynomials using coordinate transformations

I am considering multivariate polynomials of the form $$f(x,y)=x^a\,y^b\,p(x,y)^c$$ (and similarly for higher dimensions). I am trying to transform these polynomials into the generic form $$\widetilde{...
Giulio Crisanti's user avatar
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134 views

Flat locus of finite map between integral schemes is not necessarily open

Let $X,Y$ be integral Noetherian schemes. Let $f:X\to Y$ be a finite map of schemes. I recently had to show that the set of points $V\subset Y$ over which $f$ is flat is open, as is for instance ...
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Describing blow-ups locally

Let $X$ be a scheme and $U$ an open subscheme of $X$. Let $C \subseteq X$ be a closed subscheme of $X$ that is properly contained in $U$. I would like to know if we have $$\text{Bl}_{C}(X) \cong \text{...
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How to prove the blow up in a finite time of the classical solution following IBVP of semilinear heat equation?

Let $p>1$ be an even number and $\Omega\subset R^3$ be a bounded boundary $\partial\Omega$. Using an energy argument to show that the classical solution $u$ to IBVP \begin{equation} \left\{\begin{...
Wang Aliber's user avatar
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1 answer
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Lines through the origin in the blow-up of $k^n$ at the origin

Let $k$ be a field. Let $y_1,\cdots,y_n$ be homogeneous coordinates in $\mathbb{P}^{n-1}(k)$ and $x_1,\cdots,x_n$ be coordinates in $k^n$. Let $\Gamma$ be the variety defined by the $(y_1,\cdots,y_n)$-...
kiyopi's user avatar
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1 answer
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Multiplicity on strict tranform and exceptional divisor : $m_x(\hat{C}\cap E)\geq m_x(\hat{C})$ for $x\in \hat{C}\cap E$

Let $S$ be a surface, $C$ an irreducible curve on $S$ and $p\in C$ a point. Then consider the blow-up at $p$ and write $E$ the exceptional divisor. I want to show that if $x\in \hat{C}\cap E$, then $...
raisinsec's user avatar
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Strict transform of curve is smooth for finite composite of blow-ups

This is an exercise from Beauville's book called "Complex algebraic surfaces". Let $C$ be an irreducible curve on a surface $S$. We want to show that that there is a morphism $\hat{S}\to S$ ...
raisinsec's user avatar
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2 answers
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Understanding blowups at nonreduced loci

Blowing up at a reduced subscheme is geometrically clear to me: it is just like blowing up at a reduced point, and point on the exceptional divisor corresponds to different tangent lines through the ...
User X's user avatar
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Understand the Proj construction and blow-up

I have some problem understanding the Proj construction. I hope I can understand it better by the following example: Let $I\subset R$ be an ideal, and consider ${\rm Proj}(\oplus I^k)$, which is the ...
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Sheaf cohomology of blowup - reference request

I am looking for a reference for the computation of the sheaf cohomology of a blowup where things are worked out in detail. I'd like to see at least sheaf cohomology of the structure sheaf of the ...
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Orientability and blow-ups

Is it true that if I have a smooth (if you want algebraic) orientable surface (as a real manifold) then blowing up at a point will yield a non-orientable surface and vice versa (i.e. blowing up a non-...
quantum's user avatar
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Unicity in Hartshorne Corollary II.7.15

What am I missing in the following: Let $X=\mathbb{A}^2_k$ be the affine plane over an algebraically closed field $k$, and let $O$ be the origin. Let $\tilde{X}$ be the blow-up of $O$. If $\mathcal{I}$...
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irreducible components of exceptional divisors

Let $X$ be a non-singular projective variety defined over the complex numbers, and let $Y\subset X$ be a non-singular subvariety. Denote by $\phi: \tilde{X} \dashrightarrow X$ the blow-up of $X$ ...
W Sao's user avatar
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Resolution of $D_n$ singularity and exceptional divisor

Consider a $D_n$ singularity. Let's do type $D_4$ explicitly here, by defining as 0 set of $x^2+y^2z+z^3$ in $\mathbb{A}^3$. Then do a blowup, and consider the chart given by $u=x/z$, $v=y/z$ and $z$. ...
Peter Liu's user avatar
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1 answer
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About the integral closure of a DVR inside a galois extension of function field with two variables

Edit: I see my mistake now (murphy), w(X + Y) = min(w(X), w(Y)) only when the valuations are not the same. I have the following algebraic problem, which I encountred after thinking about blowups. My ...
Assaf Marzan's user avatar
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Global sections of pullback under a blowup

Let $X$ be a normal projective variety and consider a blowup of $X$ $$Y \rightarrow X.$$ It is a theorem that global sections of $\mathcal{O}_Y = f^\star \mathcal{O}_X$ biject naturally (i.e. via ...
Mathmop's user avatar
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Self intersection of exceptional divisor in blow-up of $\mathbb{P}^3$ along a curve

We work over the complex numbers. Let $X$ be the blow-up of $\mathbb{P}^3$ along a curve of genus $10$, which is the complete intersection of two cubics surfaces. The variety $X$ lives in $\mathbb{P}^...
OrdinaryAttention's user avatar
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Left exactness of the cotangent sequence

I am reading this note of Blickle on motivic integration and I am stuck at a technical point. At the beginning of the appendix, we are given a proper birational morphism $f: X' \longrightarrow X$ ...
Alexey Do's user avatar
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Blowup by reducing to affine open covers.

Let $i : X \rightarrow Y$ be a closed immersion corresponding to a finite type ideal sheaf. Let $f : B \rightarrow Y$ be a morphism of schemes. Suppose that for all affine open sets $U \subseteq Y$, $...
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Polynomial equations defining a pencil of cubics

Let $C_1, C_2$ be two distinct cubic curves in $\mathbb{P^2}$ with $F_1, F_2 \in \mathbb{C}[X,Y,Z]$ the homogeneous cubic polynomials generating the vanishing ideals of $C_1$, respectively $C_2$. We ...
user267839's user avatar
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5 votes
1 answer
280 views

Blow-up of a Pencil of Cubic Curves (Miranda's basic theory of elliptic surfaces)

In 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 Inot understand yet: Let $C_1$ be a smooth cubic ...
user267839's user avatar
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1 vote
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Dependence of strict transform on the subscheme along which we blow up

It is stated in https://stacks.math.columbia.edu/tag/080C that Note that taking the strict transform along a blowup depends on the closed subscheme used for the blowup (and not just on the morphism $...
SeparatedScheme's user avatar
2 votes
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115 views

Blowing up of a singular subvariety

I'm reading Kollár Mori Chapter 2.3. And they state the following lemma: Lemma 2.29. Let $X$ be a smooth variety and $\Delta = \sum a_iD_i$ a sum of distinct prime divisors. Let $Z\subseteq X$ be a ...
Hydrogen's user avatar
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Relation between exceptional divisor and tangent directions

Consider the hypersurface $X$ in $\mathbb{P}^2(\mathbb{C})\times \mathbb{P}^2(\mathbb{C})$ defined as the zero locus of $$ X:Z(f)= (y_1y_2+y_0^2)x_0+y_1^2x_1+y_2^2x_2=0$$ with $(x_0,x_1,x_2;y_0,y_1,...
dmenthusiast's user avatar
3 votes
1 answer
210 views

Properness of the blow ups of arithmetic schemes

In Silverman's textbook Advanced Topics in the Arithmetic of Elliptic Curves, on page 345, he defined the blow up of an arithmetic surface (or a two dimensional scheme) at the point $(\pi,x,y) = (0,0,...
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