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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 $n > 2$?

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1 Answer 1

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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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  • $\begingroup$ Sir, thank you so much for your wonderful answer. This really helps me. $\endgroup$
    – Tittu
    Commented Apr 15, 2013 at 7:46
  • $\begingroup$ Dear Tittu, it was my pleasure. $\endgroup$ Commented Apr 15, 2013 at 8:18
  • $\begingroup$ 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 $B$ $\endgroup$
    – Tittu
    Commented Apr 16, 2013 at 6:39
  • $\begingroup$ However, it is clear in the second viewpoint that the variety $B$ is projective as it is closed subset of the projective variety $\mathbb{P}^n \times \mathbb{P}^{n-1}$ $\endgroup$
    – Tittu
    Commented Apr 16, 2013 at 6:50
  • $\begingroup$ 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. $\endgroup$
    – Tittu
    Commented Apr 17, 2013 at 6:02

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