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.

Filter by
Sorted by
Tagged with
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 ...
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 : -$...
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 ...
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\...
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 &...
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 $p : Y \to X$ ...
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,.....
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 $C^{k, \alpha}$ ...
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 $C_i$?
's user avatar
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=...
's user avatar
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}_{...
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 ...
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-...
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\...
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 ...
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 $x$?
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 ...
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 ...

1
2 3 4 5 6