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.
201
questions
3
votes
0answers
46 views
algebraic de Rham cohomology of blowup of relative line
Let $$T = \mathbb{A}^1_k,\,\,\, Y = \mathbb{P}^1_T,\,\,\, X = \mathscr{B}(Y),$$ the blowup of $Y$ at a point.
I am trying to compute the de Rham cohomology $H^1_{dR}(X/T)$, but I could use some help. ...
2
votes
1answer
43 views
Projection from Blowup is Isomorphism Away from Exceptional Set
I'm following Eisenbud's description of blowing up: let $X$ be an affine algebraic variety, $R$ the coordinate ring of $X$, and let $a_1,\ldots,a_r$ generate $R$ as a $k$-algebra. Let $Y\subseteq X$ ...
1
vote
0answers
40 views
$P^2$ blow up nine points
I quote the following paragraph form Kollar-Mori on page 22:
Let $X$ be obtained from $P^2$
by blowing up at the nine
base points of a pencil of cubic curves, all of whose members are
irreducible. ...
2
votes
0answers
56 views
Why is the blow-up of 9 points an elliptic surface?
One example of elliptic fibration is obtained as follows:
Let $Z(F),Z(G)\subset\Bbb{P}^2$ be two non-singular cubics intersecting in distinct points $P_1,...,P_9$ and take the rational map
\...
1
vote
1answer
43 views
On the intersection of affine open with the blowing-up of $\Bbb C^n$ at the origin
the blowup of $\Bbb C^n$ at the origin is the subvariety of $ \Bbb P^{n-1} \times \Bbb C^n$ given by
$B = V(x_{i-1} y_j - x_{j-1} y_i \mid 1 \le i < j \le n)$.$\ \ \ $ (1)
I am interested ...
3
votes
0answers
184 views
Intersection Theory and Blow up
The following is from Fulton's Intersection Theory:
Theorem 6.7 (Blow-up Formula). Let $V$ be a $k$-dimensional subvariety of $Y$, and let $\widetilde{V} \subset \widetilde{Y}$ be the proper ...
1
vote
0answers
26 views
Symbolic Rees Algebra of an ideal in a Noetherian excellent ring
For an ideal $I$ in a commutative Noetherian ring $R$ and integer $n\ge 0$, the $n$-th symbolic power of $I$ is define as $I^{(n)}:=\cap_{P\in Ass(R/I)} \phi_P^{-1} (I^nR_P)$ , where $\phi_P : R\to ...
0
votes
1answer
34 views
Characteristic polynomial (to see which directional blow-ups have to be made)
For some positive $a\in\mathbb{R}$ let
$$
\dot{x}=ax^2-2xy,\quad \dot{y}=y^2-axy
$$
$(0,0)$ is a degenerate equilibrium and thus directional blow-ups are made.
It is said here, p. 4 below:
[...] ...
1
vote
1answer
42 views
Is there a linear PDE which has blow up solution?
I'm looking for a initial-boundary problem on a compact spatial domain of a PDE such that a unique blow up solution exists.
I don't know whether it is possible or not. Can anyone help?
3
votes
1answer
79 views
Sections of the exceptional divisor of a blowup
Let $C$ be a smooth curve in a smooth threefold $X$. Denote by $Y$ the blowup of $X$ along $C$ with exceptional divisor $E$. Then $E \rightarrow C$ is a $\mathbb{P}^1$-bundle over $C$.
Is it true ...
1
vote
0answers
81 views
Blow up the elliptic singularity $x^2+y^3+z^6=0$
It is mentioned in this question https://mathoverflow.net/questions/148826/do-there-exist-double-points-on-an-algebraic-surface-in-mathbbp-mathbbc that $X=\{x^2+y^3+z^6=0\}\subset \mathbb C^3$ defines ...
0
votes
0answers
28 views
What closure for strict transform of affine variety?
Given $\pi:X\to \mathbb{A}^2$, the blowup of $\mathbb{A}^2$ at the origin, I am trying to calculate the strict transform of $Y=\mathbb{V}(y^2-x^2(x+1))$, which has been defined as the closure of $\pi^{...
-1
votes
1answer
51 views
Self-intersection of a curve after successive blow-ups
Let $P_0,P_1,P_2\in\Bbb{P}^2$ points in general position,consider the lines $\ell_i:=\overline{P_jP_k}$ for $\{i,j,k\}=\{0,1,2\}$ and the blow-up $\pi:S\to\mathbb{P}^2$ at $P_0,P_1,P_2$.
I was told ...
1
vote
1answer
178 views
Blow up and Higher Direct Image
Let $X$ and $Y$ be smooth projective varieties and $Y \subset X$. Let $\pi : \widetilde{X} \longrightarrow X$ be the blowing up of $X$ along $Y$ with exceptional divisor $E$. Here (Direct Image by a ...
4
votes
1answer
129 views
Two way of computing blowup — which one is correct?
I came across reading Corollary 7.15 in Hartshorne's book. A special case of this corollary is the following statement
If $Y,C\subset X$ are subvarieties, $\widetilde X\to X$ is the blowup of $X$ ...
1
vote
1answer
58 views
Blow up and morphism of locally free sheaves
I would like to know if what I say below makes sense.
Let $X$, $Y$ be smooth projective varieties, $Y \subset X$ and $\pi: \widetilde{X} \longrightarrow X$ the blowing-up of $X$ along $Y$.
We know ...
0
votes
0answers
76 views
Blow up and Castelnuovo-Mumford Regularity
Let $\pi : Z = \widetilde{\mathbb{P}^{3}} \longrightarrow \mathbb{P}^{3}$ be the blowing up of $\mathbb{P}^{3}$ along an irreducible non-degenerate smooth curve $\mathcal{C}$ of degree $d$. According ...
1
vote
1answer
85 views
Just a clarification about a notation used in a question (does $I^n$ mean $I\otimes\cdots\otimes I$?)
In the following question (Direct Image by a Blow up) I have a question about the notation being used.
In this question, it is shown that $$\pi_{*}(\mathcal{O}_{\widetilde{X}}(-nE))= I_{Y/X}^{n}$$
...
2
votes
1answer
64 views
Picard group of the blow-up of $\mathbb{P}^2$ is $\mathbb{Z}\oplus\mathbb{Z}$
Let $X$ be the blow-up of $\mathbb{P}^2_\mathbb{C}$ at $P=(0:0:1)$ and $\pi:X\to\mathbb{P}^2$ the projection map. I'm trying to prove that:
$\text{Pic}(X)\simeq \mathbb{Z}\oplus\mathbb{Z}$
Here is ...
2
votes
1answer
63 views
Blowup along the fundamental locus of a rational map
Assume $f:X\dashrightarrow Y$ is a rational map between varieties, where $X$ is normal and $Y$ is complete. Then, the fundamental locus the $f$ (which means cannot extend the definition of $f$ on it), ...
0
votes
1answer
23 views
Blowup of complete along closed is complete?
Assume $X$ is a complete scheme and $B$ a closed subscheme. Is the blowup $Bl_B X$ always complete?
If $B$ is smooth, this would be clear to me. But how should I think of the case when $B$ is ...
1
vote
0answers
173 views
Blow up, ideal sheaf and direct image: can I prove this result by induction?
Let $Y\subset X$ be smooth schemes and let $\widetilde{X}$ denote the blowup of $X$ along $Y$. In this question it was shown that $$\pi_{*}\mathcal{O}_{\widetilde{X}}(-nE) = I_{Y/X}^{n}$$ for $n \geq ...
0
votes
1answer
120 views
Again, Blow up and Direct Image
In the question (Direct Image by a Blow up), follows the following statements
1) $\text{Sym}(A^{r}) \longrightarrow \bigoplus_{m \geq 0}I_{Y}^{m}$ corresponding to the closed immersion $\widetilde{X}...
1
vote
1answer
125 views
Zariski's Main Theorem and Blow up
Let $X$ and $Y$ be smooth projective schemes with $Y \subset X$. Let $\pi : \widetilde{X} \to X$ be the blow up of $X$ along $Y$ with exceptional divisor $E$.
I have seen the statement that Zariski's ...
1
vote
0answers
81 views
Direct Image, Blow up and a doubt. [duplicate]
Let $X = \mathbb{P}^{n}$ and $\pi : \widetilde{X} \longrightarrow X$ be the blow up morphism of $X$ along a subvariety $Y$ with exceptional divisor $E$.
According to the following answer in ...
2
votes
0answers
42 views
Blow up on PDE: how can I prove that $u$ is bounded in this example? [closed]
I am trying to solve the example shown in from the attached article Ball 1977.
Could you please help me prove tha $u$ is bounded in this example?
1
vote
1answer
70 views
Exceptional divisor of the blow-up of affine cone at the vertex
Let $f(x) \in \mathbb{C}[X_1,...,X_n]$ be a homogeneous polynomial in $n$ variables such that the zero locus $V$ of $f$ in $\mathbb{C}^n$ is singular only at the origin. Denote by $\pi:\widetilde{V} \...
0
votes
0answers
55 views
Picard group of strict transform
Let $X$ be a scheme (separated, noetherian, connected, over $\mathbb{Z}[\frac{1}{2}]$). If $Z\subset X$ is closed, consider the blow up $Bl_Z(X)$ of $X$ along $Z$. Then we know that the picard group ...
1
vote
1answer
103 views
Birational invariant Hodge numbers
Consider the Hodge numbers of smooth projective varieties over $\mathbb C$. I am aware of the fact that only the outer Hodge numbers (essentially $h^{p,0}$) are invariant under birational equivalence. ...
0
votes
1answer
101 views
A beginner with a question about global sections.
Let $X$ be a smooth projective scheme, $s_{1}$, $s_{2}$ global sections in $X$
and $\pi : \widetilde{X} \longrightarrow X$ be the blow up morphism of $X$ along $Y \subset X$ with exceptional divisor $...
2
votes
0answers
204 views
How can I define such a morphism?
Definition: Let $X$ be a smooth complex projective variety of dimension $n$. A holomorphic distribution of rank $k$ on $X$ is nonzero coherent subsheaf $\mathcal{F} \subsetneq T_{X}$ of generic rank $...
1
vote
1answer
30 views
When is the Blow-up of $\mathbb{P}^n$ dominant?
Let $X\subset\mathbb{P}^n$ be some algebraic variety. Lets assume that there are some homogeneous polynomials $f_0,\ldots,f_n$, all of degree $d$, that generate the vanishing ideal of $X$. When is the ...
1
vote
0answers
216 views
Blow-up, strict transform and tangent cone (Gathmann Notes, Exercise 9.22)
I'm studying Gatmann's Notes (version of 2014) https://www.mathematik.uni-kl.de/~gathmann/de/alggeom.php
I'm currently reading the Chapter 9. Birational Maps and Blowing Up. I'm trying to do exercise ...
1
vote
0answers
69 views
Blow up and finished/glued desingularization
My interest is in the "finished product", the desingularization $\tilde{X}$ of a variety (or scheme) $X$. It is widely described how to get to the different resolved charts but not how they glue ...
3
votes
1answer
310 views
Direct Image by a Blow up
Let $\pi : \widetilde{X} \longrightarrow X$ be the blow up morphism of $X$ a long of $Y \subset X$, with exceptional divisor $E$ and $\text{dim}Y > 0$, where $X$ and $Y$ are smooth projectives ...
2
votes
0answers
93 views
Sheaves morphism on Blowing up Variety.
Definition: A codimension r distribution $\mathcal{F}$ on a smooth complex manifold X is given by an exact sequence: $$0 \longrightarrow T_{\mathcal{F}} \longrightarrow T_{X} \longrightarrow N_{\...
0
votes
0answers
47 views
Pullback of a Locally Free sheaf and Blow up
Let $X$ and $Y$ be smooth projectives schemes with $Y \subset X$.
Let $f : \widetilde{X} \longrightarrow X$ be the blow up morphism along of $Y$. If $\mathcal{G}$ is a locally free sheaf on $X$, ...
1
vote
0answers
61 views
How can blow-ups be projective?
Take the blow-up $B$ of the affine plane $\mathbb{A}^2$ at the origin. I like to visualise the result as an affine plane but with a copy of $\mathbb{P}^1$ replacing the origin. Now, there's a map $B \...
0
votes
0answers
126 views
Bott's Formula and Blow up
Let $X = \mathbb{P}^{n}$ and $Z \subset X$ a smooth subvariety with $\text{dim}(Z) > 0$.
A very useful tool for calculating cohomologies in $X$ is the Bott's Formula below enunciated:
$$h^{q}(\...
0
votes
1answer
216 views
How is the degree defined in this case?
By Daniel Huybrechts, we have:
Definition: Let $E$ be a coherent sheaf of dimension $d = \text{dim}X$. The degree of $E$ is defined by: $$\text{deg}(E) = \alpha_{d-1}(E) - \text{rk}(E).\alpha_{d-1}(...
1
vote
0answers
35 views
Universal property of blowups for diagonal embedding
Corollary 7.15 of Hartshorne states: If $f:Y\to X$ is a morphism of Noetherian schemes and $\mathcal{I}$ is a coherent sheaf of ideals on $X$, then there is a diagram
$$
\begin{align*}
&\overline{...
1
vote
0answers
71 views
Reference that the blow-up of a smooth variety along a smooth subvariety is smooth.
Accoding to this post:
The blow-up of a non-singular variety along a non-singular subvariety is well-known to be non-singular
Is that difficult, or rather trivial? In the latter case, what is a ...
0
votes
1answer
89 views
Blow up of the arithmetic surface $xy = t^e$
Let $R$ be a DVR with the uniformizer $t$. Consider the arithmetic surface:
$$C_e : xy = t^e.$$
I want to repeat the computation of Liu, Algebraic Geometry, Example 3.53, Chapter 8 and compute the ...
0
votes
0answers
44 views
Subvarieties of a curve that touch near the center of a blowup (Shafevich problem 2.4.1)
Can anyone confirm or refute this proof? Also, how do I formalize the second part?
Shafevich, problem 2.4.1
Since Ī¾ is nonsingular, it's contained in a unique irreducible component of X. So, there's ...
4
votes
0answers
54 views
Are these types of blow-ups affine?
When reading Stacks Project I encountered an argument that shows some blow-up is affine. It seems that the situation can be generalised, but I am not very sure if I am missing something, so I would ...
0
votes
0answers
89 views
Blow up algebras
Consider a ring $R$, an ideal $I$ in $R$ (with generators $g_1,\ldots,g_n$). Then for some $f\in I$ the map $R\rightarrow R[x_1,\ldots,x_n]/(fx_1-g_1,\ldots,fx_n-g_n)$ should correspond to blowing up $...
5
votes
1answer
205 views
Direct image of structure sheaf under blow-up along non-singular subvariety
I'm trying to prove the following statement:
Theorem A Let $X$ be a non-singular variety over a field $k$ and let $Y \subset X$ be a smooth subvariety. Consider the blow-up $f : \widetilde X = Bl_Y(...
1
vote
0answers
77 views
Higher Direct Images of Birational Morphisms between Regular Schemes
Let $X$ be a normal, of finite type over a field of characteristic zero and regular scheme. Let $s \in X$ be a closed point. Assume we have a proper birational map
$f: Y \to X$
with the property ...
4
votes
1answer
160 views
Global Sections of Blow Up
Let $Y$ be a regular surface (therefore a proper, $2$-dimensional scheme over base field $k$) and $y \in Y$ a rational closed point (so $k(y)=k$).
Denote by $I$ the ideal sheaf corresponding to $y$ ...
1
vote
0answers
38 views
blow up of affine reducible variety at point has isomorphic total ring of fractions?
Assume $X$ is an reducible affine variety over $\mathbb{C}$ with coordinate ring $A$. Let $\pi \colon \tilde{X} \to X$ be the blowing up at a point $p \in X$ (which can be included in more than one ...
