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I have clearly understood the blowing up of $\mathbb{A}^n$ at the origin and it is the zero locus of the polynomials $x_{i}y_{j} = x_{j}y_{i}$ in the mixed product space $\mathbb{A}^n \times \mathbb{P}^{n-1}$ where $(x_1,...x_n) \in \mathbb{A}^n$ and $(y_0,...,y_{n-1})\in \mathbb{P}^{n-1}$. Please help me to understand the blowing up $\mathbb{P}^n$ at a point, say $p$.

Let me try: The blow up of a point should be a closed subset of the product space $\mathbb{P}^n \times \mathbb{P}^{n-1}$. If we blowup $\mathbb{P}^n$ in two points, then the blowup will be a closed subset of the product space $\mathbb{P}^n \times \mathbb{P}^{n-1} \times \mathbb{P}^{n-1}$. I don't know whether it is correct or not. Is it difficult to construct the blowup of $\mathbb{P}^n$ at a point explicitly when $n > 2$?

fish_monster
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Tittu
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1 Answers1

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First point of view
Consider that $\mathbb{A}^n\subset \mathbb P^n$ by identifying $(x_1,\ldots ,x_n)$ with $[1:x_1.\ldots:x_n]$ .
Then glue together the blow-up $B_0 \subset \mathbb{A}^n\times \mathbb P^{n-1}$ of $\mathbb{A}^n$ at $(0,\ldots,0)=[1:0:\ldots:0]$ and the variety $\mathbb P^n\setminus [1:0:\ldots:0] $ by identifying $((x_1,\ldots ,x_n),[x_1:\ldots:x_n])\in B_0$ with $[1:x_1:\ldots:x_n]$ whenever $(x_1,\ldots ,x_n)\neq (0,\ldots ,0)$.
The variety $B$ obtained by this gluing process is the required blow-up of $\mathbb P^n$ at $[1:0:\ldots:0]$

Second point of view
Directly describe the blow-up of $\mathbb P^n$ at $[1:0:\ldots:0]$ as the subvariety $B\subset \mathbb P^n \times\mathbb P^{n-1}$ defined by demanding for a pair $([x_0:x_1:\ldots:x_n],[y_1:\ldots:y_n])\in \mathbb P^n \times\mathbb P^{n-1}$ that the following bihomogeneous conditions of bidegree $(1,1)$ hold: $$ x_iy_j-x_jy_i =0 \quad i,j=1,\ldots, n $$

(Be sure to notice that these conditions do not involve $x_0$)

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Georges Elencwajg
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  • Sir, thank you so much for your wonderful answer. This really helps me. – Tittu Apr 15 '13 at 07:46
  • Dear Tittu, it was my pleasure. – Georges Elencwajg Apr 15 '13 at 08:18
  • I have a silly doubt: It is not clear to me that the gluing of $B_0$ with the open set $\mathbb{P}^n \backslash [1: 0 : ... :0]$ gives the projective variety $B$ – Tittu Apr 16 '13 at 06:39
  • However, it is clear in the second viewpoint that the variety $B$ is projective as it is closed subset of the projective variety $\mathbb{P}^n \times \mathbb{P}^{n-1}$ – Tittu Apr 16 '13 at 06:50
  • Sir, will it be possible for me to ask more questions on blow up? I am trying some more examples to clearly understand the concept. For example, I would like to know what will be the blowup of a point on a hypersurface in $\mathbb{P}^n$ or in general a point on any projective variety. – Tittu Apr 17 '13 at 06:02
  • Dear Tittu, you are very welcome to ask more questions, but you should ask them as new questions (by clicking on "Ask question" above, as you did for this one) rather than in the comments. – Georges Elencwajg Apr 17 '13 at 06:31
  • Do you have a reference for the second viewpoint on the blowup? I remember vaguely that one can describe the blowup of any projective variety (along an arbitrary subvariety?) in a similar way. – Michael Albanese Jul 02 '20 at 00:03
  • @Michael Albanese: Shafarevich, Volume 1, page 114. The cover has a splendid illustation (in emerald green) of a blow-up! – Georges Elencwajg Jul 02 '20 at 13:59
  • Unless I am misreading it, that only deals with the blowup of a point. Is there a similar construction for blowing up a positive dimensional subvariety? – Michael Albanese Jul 02 '20 at 14:24