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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 $P^n$.

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The 'permutohedral variety' (the toric variety obtained from the permutohedron) is a very attractive toric variety which can be obtained by blowing up all of the toric strata in $\mathbb P^n$ in increasing order of dimension (i.e. all points first, then all lines, etc.). Similarly it can be obtained from $(\mathbb P^1)^n$ by blowing up two antipodal torus-invariant points and then all toric strata that meet these two points, again taken in increasing order of dimension. I learnt this from Dhruv Ranganathan's thesis (Theorems 54 & 55). It is by no means the minimal such construction (when $n=2$ it says that the blowup of $\mathbb P^2$ in three points is isomorphic to the blowup of $(\mathbb P^1)^2$ in two points) but it is certainly a very beautiful one.

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  • $\begingroup$ This is nice. You can also consider only one point in $(\mathbb{P}^1)^n$ and only consider the toric strata in $\mathbb{P}^n$ contained in one toric hyperplane. This is minimal, at least in dimension $2$ and $3$ (see my answer below). $\endgroup$ – Jérémy Blanc Oct 23 '20 at 6:43
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Yes there is such a general statement, except for the fact that (as Nicolas suggests in his answer) you need to consider sequences of blow-ups with centers at different subspaces, not just points. The statement I know well emerges in my paper "Toric varieties of Loday's associahedra and noncommutative cohomological field theories" with Sergey Shadrin and Bruno Vallette. Let me explain the way I see the case of $\mathbb{P}^1\times\mathbb{P}^1$ and $\mathbb{P}^2$; it is easily generalisable. The definition of the varieties below goes back to work of Laura Escobar on the so called brick manifolds.

In the case of $(\mathbb{P}^1)^{n-2}$ and $\mathbb{P}^{n-2}$, I am going to construct the desired varieties as parametrising certain collections of subspaces in $\mathbb{C}^{n-1}$ whose basis I will denote $e_{1,2}$, $e_{2,3}$, ..., $e_{n-1,n}$. I denote by $G(i,j)$ the subspace spanned by $e_{k,k+1}$ with $i\le k$, $k+1\le j$.

In the particular example $n=4$, we shall consider all possible choices of subspaces $V_{2,2}$, $V_{2,3}$, and $V_{3,3}$ of $\mathbb{C}^3$ in the following diagram

where the only information we have is that $\dim(V_{i,j})=j-i+1$ and that arrows are inclusions of subspaces; in general, this will become a trapezoidal shape of height $n-1$ with the prescribed left and right slope.

If we examine this particular picture starting from the bottom row, $V_{2,2}$ is a line in the two-dimensional $G(1,3)$, $V_{3,3}$ is a line in the two-dimensional $G(2,4)$, so choices of these two are parametrized by $\mathbb{P}^1\times\mathbb{P}^1$, and the ambient $V_{2,3}$ is uniquely reconstructed as $V_{2,2}+V_{3,3}$ unless $V_{2,2}=V_{3,3}$, in which case we see that $V_{2,2}=V_{3,3}=G(1,3)\cap G(2,4)=G(2,3)$, and we have $\mathbb{P}^1$ choices, so we have to do a blow-up of $\mathbb{P}^1\times\mathbb{P}^1$ at a point.

Similarly, going from the top of the figure, we may choose $V_{2,3}\subset G(1,4)$ first, which is $\mathbb{P}^2$ of choices, and then usually $V_{2,2}$ and $V_{3,3}$ are reconstructed uniquely as $V_{2,2}=V_{2,3}\cap G(1,3)$, $V_{3,3}=V_{2,3}\cap G(2,4)$, and non-uniqueness happens if $V_{2,3}$ coincides with either $G(1,3)$ or $G(2,4)$, in each case creating $\mathbb{P}^1$ of choices, so we get the blow-up of $\mathbb{P}^2$ at two points.

In general, there will be many different blowups at different centers but the general thing remains: creating the parameter space from the bottom row will give you a sequence of blow-ups of $(\mathbb{P}^1)^{n-2}$, and creating the same space from the top row will give you a sequence of blow-ups of $\mathbb{P}^{n-2}$.

The varieties thus obtained happen to be toric, and this gives another perspective to the construction hinted at by Balazs in another answer.

It is also possible to have other examples; probably, the most famous instance of a variety that is obtained by blow-ups from both $(\mathbb{P}^1)^{n-2}$ and $\mathbb{P}^{n-2}$ is the Deligne-Mumford space of stable rational curves $\overline{\mathcal{M}}_{0,n+1}$ (the $(\mathbb{P}^1)^{n-2}$ part is very unsurprising since $n-2=n+1-3$, so the idea is more or less to take the first three points to $0,1,\infty$ and start blowing up diagonals, and for blowing up $\mathbb{P}^{n-2}$ you have to be a bit more inventive, and, for example, use the theory of wonderful compactifications of De Concini and Procesi).

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    $\begingroup$ Really nice answer ! $\endgroup$ – Nicolas Hemelsoet Oct 19 '20 at 8:32
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    $\begingroup$ Very nice. Can one use this though to get the Poincare polynomial/motivic identity implicit in Nicolas' approach? Does one get some interesting combinatorial identity? $\endgroup$ – Balazs Oct 19 '20 at 11:16
  • $\begingroup$ @Balazs : I haven't done that calculation explicitly but yes, it can definitely be done. The coefficients of the Poincaré polynomial form, of course, the h-vector of the corresponding Stasheff polytope, so the total dimension of the cohomology is given by the appropriate Catalan number, and the individual Betti numbers are the "Narayana numbers", so I assumed that all combinatorial identities one will obtain will be known to experts in combinatorics, and did not spend too much time playing with them! $\endgroup$ – Vladimir Dotsenko Oct 19 '20 at 12:41
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$(P^1)^n$ has Poincaré polynomial $(1+t^2)^n$, and $P^n$ has Poincaré polynomial $1+t^2+ \dots + t^{2n}$. If $X$ is of dimension $n$ and $B_pX$ the blow-up at a point $p \in X$ then $p_{B_pX}(t) = p_X(t) + t^2 + \dots + t^{2n-2}$.

If $B_r(P^1)^n$ is the blow-up of $(P^1)^n$ at $r$ points, and $B_s(P^n)$ the blow-up of $P^n$ at $s$ points, then equality of Poincaré polynomial implies that $$(1+t^2)^n + r(t^2 + \dots + t^{2n-2}) = 1 + t^2 + \dots + t^{2n} + s(t^2 + \dots + t^{2n-2})$$ So we see that $r+n-1=s+1$, $r + \binom{n}{2}= s+1$, etc... and we easily see that this system that has no solutions if $n \geq 3$.

Of course, a less naive generalisation involving blowing-up at linear subspace could still work.

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[This is a correction of an earlier answer, and is currently incomplete; thanks to Nicolas for pointing out that what I had written was totally wrong.]

Both varieties are rational. The obvious birational map between them factors into a composite of smooth blowups and blowdowns. As they are toric varieties, they carry a dense torus action, and this factorisation is also equivariant. We can use toric geometry to try and see what happens (see Fulton's book Introduction to Toric Varieties for this language; the first few sections will suffice for what I am saying).

Consider the lattice $N={\mathbb Z}^n$ with standard basis $e_1, \ldots, e_n$. Then ${\mathbb P^n}$ corresponds to the toric variety given by the fan $\Sigma_1$ spanned by the lattice vectors $\{e_1, \ldots, e_n, -\sum_i e_i\}$, and all $n$-subsets of these vectors spanning top-dimensional cones.

The product ${\mathbb P^1}\times\ldots\times{\mathbb P^1}$ corresponds to the toric variety associated to the fan $\Sigma_2$ spanned by $\{\pm e_1, \ldots, \pm e_n\}$, and top-dimensional cones consising of points with all coordinates of fixed sign (the analogue of coordinate quadrants for $n=2$).

Consider the fan $\widetilde\Sigma_1$ obtained by inserting the lattice vector $-e_i$ into the interior of each maximal-dimensional cone of $\Sigma_1$ except the positive cone. Unless I am still making a mistake, it seems to me that the corresponding toric morphism $\widetilde{\mathbb P}_1\to {\mathbb P}^n$ is the blowup of $n$ disjoint points.

On the other hand, consider the fan $\widetilde\Sigma_2$ obtained by subdividing the cone in $\Sigma_2$ spanned by all the points with nonpositive coordinates using the lattice vector $-\sum_i e_i$. We get a toric one-point blowup $\widetilde{\mathbb P}_2\to {\mathbb P^1}\times\ldots\times{\mathbb P^1}$.

The two toric varieties $\widetilde{\mathbb P}_1, \widetilde{\mathbb P}_2$ are defined using fans on the same set of 1-simplices $\{\pm e_1, \ldots, \pm e_n, -\sum_i e_i\}$, but the triangulations (sets of maximal cones) are totally different for $n>2$. The birational map $\widetilde{\mathbb P}_1\dashrightarrow \widetilde{\mathbb P}_2$ can be understood as a composite of small birational modifications, corresponding to changes of triangulation between these two fans. I cannot see a systematic way to describe what happens; one can understand small $n$ explicitly.

Alternatively, as Nicolas also suggests, we should also be able to do further blowups/blowdowns to match the triangulations also, but I don't see a systematic way to do so.

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  • $\begingroup$ (I deleted my now obsolete comment) Blowing-up along a $P^k$ add $t^{2k+2} + \dots + t^{2n-2-2k}$ to the Poincaré polynomial, so we could actually compute the exact values of how many linear subspaces of various dimensions we should blow-up in $\Bbb P^n$ to get at least there right number. The solution is probably as you said to understand how to relate the triangulations you introduced. $\endgroup$ – Nicolas Hemelsoet Oct 19 '20 at 7:18
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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).

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    $\begingroup$ What is a pseudo-isomorphism? $\endgroup$ – R. van Dobben de Bruyn Oct 22 '20 at 22:37
  • $\begingroup$ A pseudo-isomorphism $X\dashrightarrow Y$ is an isomorphism $U\to V$ where $U\subset X$ is open, $V\subset Y$ is open and the closed subsets $X\setminus U$ and $Y\setminus V$ are of codimension at least $2$. $\endgroup$ – Jérémy Blanc Oct 23 '20 at 6:39

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