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.
298
questions
1
vote
1
answer
70
views
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 $...
2
votes
0
answers
45
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 ...
- 539
0
votes
1
answer
82
views
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,...
0
votes
0
answers
18
views
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 ...
- 7,603
3
votes
1
answer
81
views
Properness of the blow ups of arithmetic schemes by Silverman
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,...
- 43
2
votes
0
answers
30
views
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 :
-$...
- 93
2
votes
1
answer
74
views
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 ...
- 23
1
vote
0
answers
22
views
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 &...
- 625
4
votes
1
answer
78
views
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 ...
- 569
0
votes
1
answer
58
views
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 ...
- 87
0
votes
1
answer
75
views
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 ...
0
votes
0
answers
46
views
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 ...
0
votes
0
answers
62
views
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 ...
2
votes
0
answers
64
views
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\...
- 595
0
votes
0
answers
25
views
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/...
- 2,076
2
votes
0
answers
69
views
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 ...
0
votes
1
answer
59
views
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 ...
- 2,145
1
vote
1
answer
124
views
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{...
0
votes
1
answer
145
views
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 ...
1
vote
1
answer
139
views
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}$$
be a commutative square with the two horizontal maps closed embeddings ...
- 97
1
vote
1
answer
54
views
Resolving the singularity arising from $\vec{x} \cdot \vec{x} = 0$
I'm just learning blowups and resolutions of singularities and I have been unable to find a clear and concise resource on how to resolve singularities in general. I understand that the concept of &...
- 21
2
votes
1
answer
58
views
Differential inequality and blow-up: explanation of part of a proposition
I wanted to understand the end of the proof of this proposition, it is part of this paper: ANALYSIS OF A CONVECTIVE REACTION-DIFFUSION EQUATION II*, by H.A. LEVINE, L. E. PAYNE, P. E. SACKS, and B. ...
- 2,653
2
votes
1
answer
182
views
Picard group of blowup of smooth variety
If $X$ is a smooth irreducible variety and $Y$ is the blowup of $X$ at a point $p$ then
Prove that $Pic(Y) = Pic(X)+$$\mathbb Z$
I was thinking about using the fact that the blowup map $p : Y \to X$ ...
- 2,096
1
vote
0
answers
57
views
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 ...
- 109
0
votes
0
answers
32
views
Macaulay2 how to change grading of ReesAlgebra for computing HilbertSeries
I am trying to compute the Hilbert-Samuel polynomials of some examples in Macaualy2.
In my toy example, $R = \mathbb{Q}[x,y]/(x^2 - y^3)$ and I am considering the ideal $I = (x,y)$. Then I am looking ...
- 1,344
1
vote
0
answers
39
views
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, ...
- 115
1
vote
0
answers
66
views
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 ...
- 1,515
0
votes
0
answers
28
views
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 ...
- 2,364
0
votes
0
answers
61
views
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,.....
- 831
1
vote
1
answer
109
views
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 $...
- 8,965
2
votes
1
answer
79
views
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 $X$ ...
- 35
3
votes
0
answers
53
views
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 $C^{k, \alpha}$ ...
- 111
3
votes
1
answer
220
views
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
3
votes
1
answer
177
views
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 ...
- 13.1k
2
votes
0
answers
35
views
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 $$\text{Bl}_0 K^2=\left\{ ((x_1,x_2),[y_1,y_2])\in K^2\times \mathbb{P}^1_K \lvert x_1y_2=...
user948537
1
vote
2
answers
78
views
Linking Blow-ups
I would like to know when we can "zig-zag-connect" blow-ups of the same base space, by which I mean the following: Let $X$ be a $k$-scheme and $Z_1,Z_2$ two closed subschemes and $\text{Bl}_{...
- 1,928
2
votes
0
answers
97
views
Blow-up, Exercise 22.2.A, Vakil.
Vakil (FOAG) defines the blow-up $X \hookrightarrow Y$ (closed subscheme corresponding to a finite type quasicoherent sheaf of ideals) to be Cartesian diagram
$$\require{AMScd}
\begin{CD}
E_X Y @>&...
- 2,116
1
vote
1
answer
59
views
Finding infinitely many cubics on $\Bbb{P}^2$ whose strict transforms are exceptional curves
Let $\Bbb{P}^2$ be the projective space over $\Bbb{C}$.
It is a well known fact that the blowup of $\epsilon:S\to\Bbb{P}^2$ at $9$ points $P_1,...,P_9$ in general position generates infinitely many ...
- 8,965
3
votes
1
answer
122
views
Hartshorne Proposition II.7.13a
Let $X$ be a Noetherian scheme, $\mathscr{I}$ be a coherent sheaf of ideals and let $\pi:\operatorname{Bl}_Y X \rightarrow X$ be the blow up of $X$ along the closed subscheme $Y$ corresponding to $\...
- 423
8
votes
0
answers
75
views
Where do toric varieties appear naturally?
I'm reading Fulton's book. There's an awesome theorem that classifies all smooth toric surfaces as blowups at points starting from either $P^2$ or some Hirzebruch surface.
I want to be more excited ...
- 1,824
1
vote
1
answer
247
views
Resolution of singularities of analytic spaces
It seems to me that the following resolution of singularities theorem (or a modification) is known to specialists but I have trouble finding references.
Let $X$ be a complex analytic space, then there ...
- 336
0
votes
0
answers
44
views
Minimal embedding for blowing ups
Let us consider the following specific problem for blowing-ups.
Let $n$ be a large positive integer. Let $X\subset \mathbb P^n$ be a smooth sub variety of codimension $>1$. Denote $Y$ the blowing-...
- 336
0
votes
1
answer
66
views
Exceptional divisor of the Lefschetz pencil
I am reading a proof due to Zucker showing that integral Hodge conjecture holds for cubic fourfolds.
Let $X\subset \mathbb{CP}^5$ be a cubic fourfold. We consider a Lefschetz pencil $(Y_t)_{t\in\...
- 336
1
vote
0
answers
40
views
Is the Kempf-Laksov-resolution of a Gorenstein single-condition Schubert variety a blowup?
Let $Gr(k,V)$ be the Grassmann bundle of $k$-dimensional subspaces of an $n$-dimensional vector space $V$ equipped with a full flag $0=E_0\subset E_1 \subset \ldots \subset E_{n-1}\subset E_n=V$. ...
- 145
2
votes
1
answer
87
views
Counting geometrical constraints by blowing-up
Let $X$ be a smooth projective variety, and let $|D|$ be a complete linear system of divisors on $X$. For a given codimension-two locus $S$ within $X$, a natural question is: what is the dimension of ...
- 1,389
2
votes
0
answers
72
views
K3 surface is not the blow up of any other smooth surface.
This is the exercise 2.5.5 in the book 'Complex Geometry' by Huybrechts:
Let $X$ be a K3 surface, i.e. $X$ is a compact complex surface with $K_X\cong\mathscr{O}_X$ and $h^1(X,\mathscr{O}_X)=0$. Show ...
- 411
2
votes
0
answers
34
views
Can a sequence of blowups be written as one blowup with multiple exceptional divisors (and vice versa)?
Consider this example:
I have 3 divisors $D_1, D_2, D_3$ in a variety $X$, let's say for simplicity that they have simple normal crossing. Now let's say I blow up the intersection $Z_1 = D_2 \cap D_3$...
- 21
1
vote
1
answer
292
views
Blow-up of $\mathbb{P}^1 \times \mathbb{P}^1$ at a point.
Let $x:= (0,0) \in \mathbb{P}^1 \times \mathbb{P}^1$. What is the blow-up of $\mathbb{P}^1 \times \mathbb{P}^1$ at $x$?
- 408
6
votes
0
answers
153
views
depicting blowups
Is there a way to make nice images of blowups?
For instance you can always find a picture of the blowup of $y^2 = x^2 +x^3$
for example, https://www.math.purdue.edu/~arapura/graph/nodal.html
But I ...
- 313
0
votes
0
answers
95
views
Understanding blowups
I'm looking for a easy to work with definition of a blowup with an example/someone to check my undestanding of a blowup.
I understand that blowups are a method that we use to deal with singularites of ...
- 313