# 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
Filter by
Sorted by
Tagged with
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. ...
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 $Y\subseteq X$ ...
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. ...
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 \...
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 ...
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 ...
26 views

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 ...
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 ...
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 $X$ ...
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 ...
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 ...
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 $$\pi_{*}(\mathcal{O}_{\widetilde{X}}(-nE))= I_{Y/X}^{n}$$ ...
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 ...
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), ...
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 ...
173 views

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 ...
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 ...
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?
70 views