TITLE: Blow-up of projective variety $P^1 \times P^1..... \times P^1$ ($n$ times) and blow-up of $P^n$
QUESTION [9 upvotes]: It is known that the blow-up of $P^1 \times P^1$ at a point is isomorphic to the blow-up of $P^2$ at two points. I'm wondering if there is any general statement for the blow-up of $P^1 \times P^1 \times.... \times  P^1$ ($n$ times) and of $P^n$.

REPLY [3 votes]: The blow-up of $(\mathbb{P}^1)^n$ at one point is always pseudo-isomorphic to the blow-up of $\mathbb{P}^n$ at $n$ points in general position.
To see this, let us consider, for each $n\ge 2$, the toric birational map $\tau\colon (\mathbb{P}^1)^n\dashrightarrow (\mathbb{P}^n)$ given by $$([x_1:1],[x_2:1],\ldots,[x_n:1])\mapsto [x_1:\cdots:x_n:1]$$
and which restricts then to an isomorphism on the affine space $\mathbb{A}^n$.
You can write $\tau$ as
$$ ([x_1:y_1],[x_2:y_2],\ldots,[x_n:y_n])\mapsto [x_1y_2\cdots y_n:y_1x_2y_3\cdots y_n:\cdots:y_1\cdots y_{n-2}x_{n-1}y_n:y_1\cdots y_{n-1}x_n:y_1y_2\cdots y_n].$$
and observe that it is the blow-up of the point $([1:0],\ldots,[1:0])$ followed by a pseudo-isomorphism (isomorphism if $n=2$, flop of three curves if $n=3$, ...) and then the contraction of the strict transforms of the hyperplanes given by $y_1=0,\ldots,y_n=0$ onto the toric points of $\mathbb{P}^n$ that are contained in the hyperplane at infinity.
You can also see the pseudo-isomorphism as blow-ups and blow-downs if you prefer (for instance, in dimension $3$ it is only $3$ disjoint Atiyah flops: blow-ups of three lines and contractions of three lines).