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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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-...
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When is Total Transform under Blow-up reduced?
Question
Let $X$ be a reduced scheme, let $Z \subset Y$ be reduced closed subschemes. Denote by $\widetilde{X} \to X$ the blow-up of $X$ at $Z$.
What conditions can guarantee that $Y \times_X \...
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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 $X$ ...
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Birational maps from smooth rational minimal projective surfaces to Hirzebruch surfaces - (Beauville Exercise III.24-7)
Let $\phi\colon S\dashrightarrow\mathbb{F}_n$ be a birational map from a smooth minimal rational projective surface to the Hirzebruch surface $\mathbb{F}_n:=\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}_{\...
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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 $z$. ...
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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 ...
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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 ...
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Points in a blowing up determined by an infinitesimal curve not in the centre.
Let $\Bbbk$ be a field. Let $A$ be a $\Bbbk$-algebra (commutative and unital). All ring maps are $\Bbbk$-algebra homomorphisms. Let $I\subseteq A$ be an ideal. Let $X:=\mathrm{Spec}(A)$ and $Z:=\...
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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}^...
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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 $f: X' \longrightarrow X$ ...
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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 ...
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By hand calculation of a Blowup of a Pencil of curves
I have one more question about the Example (I.5.1) page 7 from
Rick Miranda's the basic theory of elliptic surfaces:
Let $C_1$ be a smooth cubic curve in $\mathbb{P^2}$ and let $C_2$
be any other ...
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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 ...
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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 $...
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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 ...
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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,...
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Doubt about the multiplicity of a point along a curve that lives in a surface
Suppose to have the following surface $S$ in $\mathbb{P}^3(x_0,x_1,x_2,x_3)$:
$$ x_3^6=x_0^6+x_1^6$$
Now consider the divisor $D:=(x_3)$ on $S$ and the point $p:=(1:e^{\frac{\pi i }{6}}:0:0)$ . Which ...
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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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Picard groups of blow-up
Let $X$ be a compact complex manifold and $Y$ a complex sub manifold of codimension $\ge 2$. If $\pi : X_{Y} \mapsto X$ is the blow-up of $X$ along $Y$, do you have any references for this result :
-$...
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How to show $x_0^2+x_1^2+x_2^2=0 \subset \mathbb{CP}^2 \iff \mathbb{CP}^1$
I am currently trying to blow-up an $A_n$ singularity defined by the hypersurface equation:
\begin{equation}
z_1^2+z_2^2+z_3^{n+1}=0 \subset \mathbb{C}^3
\end{equation}
Let $x_i, i=0,1,2$ denote the ...
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Extendability of a solution of a IVP
Let $\Omega \subset \mathbb{R} \times \mathbb{R}^n$ be an open (non-empty) set. Let $f \; : \; \Omega \to \mathbb{R}^n$ be a continuos function such that
$$\forall (t,x) \in \Omega \;\;\exists \delta &...
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Blowing Up the Indeterminancy Locus - Why is this sheaf invertible?
The following is explained in Hartshorne, chapter 2.7. I will be considering varieties instead of schemes. Let $X$ be a variety over $k$, and let $L$ be a line bundle on $X$. Let $s_0,\dots,s_n$ be ...
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Desingularization of the standard Cremona involution of $\mathbb P^2$.
Consider the birational map $\chi$ given by the blow up of the points $(1:0:0)$, $(0:1:0)$, and $(0:0:1)$ of $\mathbb P^2$, followed by the contraction of the strict transforms of the three lines ...
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Blow-up of 3-dimensional affine space through the line $x_1=x_2=0$.
I'm trying to understand blow-ups. In the Gathmann's notes there is an exercise:
Let $\widetilde{\mathbb A^3}$ be the blow-up of $\mathbb A^3$ at the
line $V(x_1,x_2)\equiv \mathbb A^1$. When the ...
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Is the Exceptional Divisor of a Blow-up of a Variety at a point isomorphic to projective space?
Suppose $X$ is a variety (irreducible quasi-projective algebraic set) of dimension $d$, $x\in X$, and $\pi:\tilde X\to X$ the blow-up of $X$ at $x$. Then is it true in general that the exceptional ...
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Exceptional divisor of blow-up in a non-rational point
Let $X$ be an prjective variety over $k$ and $Spec(L) \to X$ be a point of degree $n$ on $X$. Is there any description of $Bl_{Spec(L)}X$ in terms of $L$? Is there any connection between them in the ...
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Hartshorne Theorem II.7.17 - Why is $\mathscr{I} \to \mathscr{M}^{-dn}f_*\mathscr{L}^d$ injective?
I'm reading the proof of the following theorem from Hartshorne (Theorem II.7.17), which says the following:
Let $Z$ be a variety and let $X$ be a quasi projective variety, both over $k$. Suppose $f:Z\...
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Resolution of $(f=0)$ where $f(x,y,z)=x^ay+y^az+\omega z^ax$
Let $u\geq 2$ be an integer, $a=3u-1$, $\omega=e^{2\pi i /3u}$, and let $f$ be the polynomial $f(x,y,z)=x^ay+y^az+\omega z^ax$.
In Example 23 of https://www.intlpress.com/site/pub/files/_fulltext/...
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The Jacobian ideal under Blow up
Let $X$ be a singular hypersurface in $\mathbb{C}^n$ defined as the zero locus of $f(x_1, \dots, x_n)$ and denote $Jac(f)=\langle \partial_1(f), \dots, \partial_n(f) \rangle$, where $\partial_i(f)$ is ...
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Resolution of ADE- singularities: The second blow-up for the A_2 singularity
I try to resolve the surface singularity $X\subset \mathbb{A}^3$ of type $A_2$. It is defined by $x^2+y^2+z^3=0$. I expect to need at least two blow-ups. But after the first blow-up of $X$ the strict ...
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Hartshorne Exercise II.7.7(c) Studying a particular blow-up and ruled surface.
For reference, see the image below for the exercise. I have been able to solve (a) and (b), but (c) is giving me trouble.
First off, let me assume that $P=(0,0,1)$ so that the linear system $\mathfrak{...
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Hartshorne Exercise II.8.5(b) computing the canonical bundle of a blowup
Let $X$ be a nonsingular variety with $Y$ of codimension $r\geq 2$ nonsingular subvariety. Let $\pi:\widetilde{X}\rightarrow X$ be the blow of $X$ along $Y$ and $Y'=\pi^{-1}(Y)$ the exceptional ...
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Proof of Blow Up Closure Lemma (22.2.G Vakil)
To set up the problem, let
$$\require{AMScd}
\begin{CD}
W @>>> Z\\
@VVV @VVV \\
X @>>> Y
\end{CD}$$
be a commutative square with the two horizontal maps closed embeddings ...
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Resolving the singularity arising from $\vec{x} \cdot \vec{x} = 0$
I'm just learning blowups and resolutions of singularities and I have been unable to find a clear and concise resource on how to resolve singularities in general. I understand that the concept of &...
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Differential inequality and blow-up: explanation of part of a proposition
I wanted to understand the end of the proof of this proposition, it is part of this paper: ANALYSIS OF A CONVECTIVE REACTION-DIFFUSION EQUATION II*, by H.A. LEVINE, L. E. PAYNE, P. E. SACKS, and B. ...
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Picard group of blowup of smooth variety
If $X$ is a smooth irreducible variety and $Y$ is the blowup of $X$ at a point $p$ then
Prove that $Pic(Y) = Pic(X)+$$\mathbb Z$
I was thinking about using the fact that the blowup map $p : Y \to X$ ...
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An explicit example of the blowup of an algebraic curve
Consider the curve $x^3+y^4=0$ in $\mathbb{A}^2$, I am trying to retrieve the Dynkin diagram of $E_6$ by consecutive blowups at the origin, however I run into issues after performing just one of these ...
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Total and strict transforms of blowups
I have a rather simple question about the nature of blowups:
Let's say I have $\pi: X \rightarrow Y$, which is the blowup of a center $Z$ in $Y$. Denote the exceptional divisor by $E$.
Is it correct, ...
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Blow up points on real projective plane
What is the genus of the blow-up at $3$ non-colinear points of $\mathbb{R} \mathbb{P}^2$? This question occurred to me as I was trying to eliminate the sigular points at the vertices of a given ...
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Sheaf of divisor divisor $rH+sE$ on blowup $\operatorname{Bl}_0(\mathbb{P}^2)$ has basepoints but is generated by global sections?
I work with toric varieties and took a look at the blowup $\pi:X=\operatorname{Bl}_0(\mathbb{P}^2)\to \mathbb{P}^2$ ($0=(0:0:1)$) with points represented as $(x_0:x_1:x_2),(y_0:y_1)$. Technically we ...
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Lift rational map to the Blow-up by explicite construction
We work over complex numbers. Let $X \subset \mathbb{P}^n$ a complex projective subvariety
associated to homogeneous ideal $I(X) \subset \mathbb{C}[X_0,...,X_n]$.
Assume $X$ contains point $p:= (1,0,.....
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How many blowups do we need to make a pencil base point free?
Let $\Bbb{P}^2$ be the projective plane over $\Bbb{C}$.
Take a pencil of curves of degree $d$ on $\Bbb{P}^2$ given by a dominant rational map $\phi:\Bbb{P}^2\dashrightarrow \Bbb{P}^1$.
The pencil has $...
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Arrow-reversing Proj and blow-up
Let $X$ be a normal projective variety over the field of complex numbers. Let $Y$ be a subvariety of $X$, and let $I_Y$ be the ideal sheaf of $Y$. From what I know, I can define the blow-up of $X$ ...
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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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Can blow-up of a surface be a product of two curves? [closed]
Is there any smooth projective surface $S$ over $k=\bar{k}$, such that the blow up $\tilde{S}$ along some point $x\in S$ can be written as $\tilde{S}=C_1\times C_2$ for two curves $C_i$?
user949601
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Criteria for finite time blow up for two simple ODEs
I've got two simple questions on criteria for a finite time blow up of solutions of two simple ODEs:
1: If $u$ is a solution of $u'=f(u)\ge0$, $u(0)=u_0$, how do we see that $u$ blows up in finite ...
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Blow-up of origin of $K^2$ with $K$ a non-archimedian local field
Let $K$ be a non-Archimedian local field. We can define $\mathbb{P}^1_K$ in the standard way and the blow up $$\text{Bl}_0 K^2=\left\{ ((x_1,x_2),[y_1,y_2])\in K^2\times \mathbb{P}^1_K \lvert x_1y_2=...
user948537
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Linking Blow-ups
I would like to know when we can "zig-zag-connect" blow-ups of the same base space, by which I mean the following: Let $X$ be a $k$-scheme and $Z_1,Z_2$ two closed subschemes and $\text{Bl}_{...
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