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.
359 questions
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Rigidity of a subscheme
I would like to ask whether there is a connection between the rigidity of an exceptional curve described by $E^2=-1$ and the concept 'rigidity' in the deformation theory. Roughly I am looking for some ...
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If we blow up $X^{\mathbb{G}_m}$ every time, is $X^{\mathbb{G}_m}$ eventually a divisor?
Consider over $\mathbb{C}$. Consider a $\mathbb{G}_m$-variety $X$. Then $X^{\mathbb{G}_m}\hookrightarrow X$ is a closed subscheme (multiple components sometimes). We can blow $X$ along $X^{\mathbb{G}...
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How to correctly calculate the intersection number of the exceptional line with the canonical divisor of the blowup?
Let $X$ be a projective subvariety in $\Bbb P^{3}$ which is given by the equation $xyw-z^{3}=0$. Let $\widetilde{X} \rightarrow X$ be the blow up at the point $q :[x:y:z:w]=[0:0:0:1]$. There are two ...
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Blowup not smooth
I came across the following problem:
Let $k=F_p(t)$. Take the closed subscheme $S=V(x^p-tz,y)$ of $\mathbb{P}^2=\text{Proj }k[x,y,z]$. Show the blowup of $S$ is not smooth.
I suppose the first point ...
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Smooth circle bundles over blowup of weighted proective space
Consider the toric variety $X$ given by the (possibly weighted) blowup of the weighted $\mathbb{CP}_{[a_0, a_1, a_2]}$ in two of the three fixed points. If I am not mistaken, the second cohomology of $...
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Universal property of blow-up
I am reading the Gortz's Wedhorn, Algebraic Geometry, p.414~p.415 ( dealing with construction of blow-up and its universal property ) and stuck at some (final) statement. Although the universal ...
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(Blow-up) Resolution of singularities and arithmetic genus
Suppose $X$ is a $k=\overline{k}$-variety with a node at a closed point $p$. Let $\tilde{X}$ be the blow-up of $X$ at $p$.
In Exercise 28.3.E, Vakil asks to show that there is an exact sequence $0\to \...
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Reducedness of blowup implies reducedness of original scheme?
Let $X$ be a scheme, and let $X'\to X$ be a birational morphism by blowing up an ideal of $X$. Suppose $X'$ is reduced (or even smooth). Is it possible to conclude that $X$ is reduced as well?
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Relationship between covers of base of blow-up and blow-up?
Suppose we have the following set-up
$$\require{AMScd}
\begin{CD}
Y @<{\exists ?}<< \widetilde{\mathrm{Bl}_Z X};\\
@V{f}VV @V{g}VV \\
X @<{\pi}<< \mathrm{Bl}_Z X;
\end{CD}$$
where $f$...
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Blow up's and contracting Exceptional divisor along a relevant chart (Elementary transformations)
Sorry this is a question due to lack of understanding of blow ups and birationally geometry in general.
Consider the hypersurface in $\mathbb{A}^3$ given by:
$$X =xy - z^2 = 0,$$
which has a ...
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Use blow-up to extend a morphism
I've learnt about blow-ups in a basic algebraic geometry course which only deals with varieties (integral, finite type, separated). It looks a bit different from the usual definition using schemes.
...
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Blow-up of irreducible/reduced is irreducible/reduced
Let $Y$ be some scheme and $X$ be a closed subscheme of $Y$ corresponding to a finite type quasicoherent sheaf of ideals. I was trying to do Exercise 22.2.C of Vakil's "The Rising Sea":
&...
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The restriction of a blow up to a subscheme is the the blow up of this subscheme at the intersection of it and the center.
Suppose $Z \subset X$ is a closed subscheme and $Y \subset X$ is another closed subscheme. Let $\mu : \mathrm{Bl}_{Z} (X) \to X$ be the blow-up of $X$ at Z. Then is $\mu|_{Y} : \mu^{-1}(Y) \to Y$ the ...
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Surjectivity of the pushforward of a blowing-up morphism
Given a closed immersion $Z\rightarrow X$ and considering the blowing-up of $X$ at $Z$ and we get $\widetilde{X}$. I'm wondering is the proper pushforward on etale cohomology induced by the blowing-up ...
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Computing $K_{X'/\mathbb{A}^2}$
Let $k$ be an algebraically closed field of characteristic zero, $X=\mathbb{A}^2_k$ and $X'=\operatorname{Bl}_{(x,y)} X$. I'm looking to compute $K_{X'/X}$.
Note that $$X'=\operatorname{Proj} k[x,y,...
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Does maximal contact hypersurface (m.c.h.) exist for any curve over a field of arbitrary characteristic, and if so, for what definition of m.c.h.?
Does a maximal contact hypersurface always exist for any curve over a field of arbitrary characteristic, and if so, for what definition of maximal contact hypersurface?
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Projection Morphism of Blowup
I'm currently reading the article A Short Course on Geometric Motivic Integration, by Manuel Blickle. In his proof of Theorem 3.3, the author considers the following.
Let $X'=\operatorname{Bl}_{0}\...
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Are projective birational morphisms blowups?
This question seems to claim that a projective birational maps $f: X \to Y$ between varieties $X,Y$ over $\mathbb C$ is a blow-up. What is a reference for that?
According to Wikipedia,
The "Weak ...
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Confusion about stability of PDEs.
I have been reading about Stability Theory and have been left with some questions at is seems to me that some of its notions are not very well-defined or at least inconsistently used.
Consider the ...
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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 ...
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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 ...
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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 ...
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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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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)\...
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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,
$$\theta_t = \...
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bigness of divisors under blow-up of finitely many points of the plane
Let $S\subset\mathbb{P}^2$ be a finite set of points in the plane, consider the blow-up of $\mathbb{P}^2$ along $S$:
$$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].$$
The number $x_0$ can ...
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Confusion about line bundles and the intersection product
Let $X = \mathbb{A}_{\mathbb{C}}^{2}$ and let $Y$ be the blowup of $X$ at the origin. Let $E \cong \mathbb{P}^{1}$ the exceptional divisor.
I think that we have a canonical inclusion $\mathcal{O}_{Y} \...
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Liu, Proposition 8.1.12: inverse image of blown-up ideal sheaf is invertible
I am struggling with Proposition 8.1.12e of Liu's Algebraic geometry and arithmetic curves.
The setup is as follows: let $X=\text{Spec}A$, $I=(f_1,\dots,f_n)\subseteq A$ an ideal, and let
$$\tilde A=\...
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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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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{...
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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 $(y_1,\cdots,y_n)$-...
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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 $...
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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 $\hat{S}\to S$ ...
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2
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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 ...
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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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