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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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. ...
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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 $Y\subseteq X$ ...
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$P^2$ 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. ...
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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 \...
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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 ...
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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 ...
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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 ...
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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: [...] ...
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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?
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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 ...
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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 ...
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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^{...
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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 ...
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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 ...
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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 $X$ ...
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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 ...
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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 ...
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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 $$\pi_{*}(\mathcal{O}_{\widetilde{X}}(-nE))= I_{Y/X}^{n}$$ ...
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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 ...
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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), ...
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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 ...
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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 ...
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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}...
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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 ...
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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 ...
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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?
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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} \...
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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 ...
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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. ...
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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 $...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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_{\...
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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 $X$, ...
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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 \...
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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}(\...
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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}(...
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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{...
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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 ...
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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 ...
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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 ξ is nonsingular, it's contained in a unique irreducible component of X. So, there's ...
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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 ...
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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 $...
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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(...
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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 ...
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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 $y$ ...
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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 ...

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