What happens to the cohomology ring after a "flip-flop"?

Question

I've been trying to understand what happens to the cohomology ring (say with coefficients in $\mathbb{R}$) of a smooth complex projective manifold after blowing up along a smooth complex submanifold. I would first just like to double check my facts, and then I have some other general, perhaps naive, questions about blowing down.

For blowing up, I would just first like to know if I've got my story straight. The following is what I've gathered mostly from Griffiths' and Harris' book (pgs 605-608):

Let $Y\subset X$ be a complex submanifold of codimension $n$ embedded in a complex projective manifold of dimension $d$, let $\tilde{Y}\subset\tilde{X}$ be the exceptional divisor sitting inside the blow up manifold, and let $\beta\colon\tilde{X}\rightarrow X$ be the blow down map.

Then $\tilde{Y}$ is isomorphic to $\mathbb{P}\left(\mathcal{N}_{Y/X}\right)$, the projectivization of the normal bundle of $Y\subset X$, and the restriction of blow down map $\beta_0\colon\tilde{Y}\rightarrow Y$ is the usual projection map $\mathbb{P}\left(\mathcal{N}_{Y/X}\right) \rightarrow Y$. In particular, the fibers of $\beta_0$ are copies of $\mathbb{CP}^{n-1}$, and there is a class $z\in H^2(\tilde{Y})$ that restricts to each fiber to give minus the hyperplane class $H^2(\mathbb{CP}^{n-1})$--this can be taken as the first Chern class of the normal bundle to $\tilde{Y}\subset\tilde{X}$. Thus by the Leray-Hirsch theorem we get an $H(Y)$-module decomposition $$H(\tilde{Y})\cong H(Y)\oplus z H(Y)\oplus\cdots z^{n-1}H(Y).$$

If we take the Poincare dual of the fundamental class of the exceptional divisor, say $\tau\in H^2(\tilde{X})$, then we get an $H(X)$-module decomposition $$H(\tilde{X})\cong H(X)\oplus \tau H(Y)\oplus\cdots\tau^{n-1} H(Y)$$ and the restriction of $\tau$ to $\tilde{Y}$ is equal to $z$.

Question 0: Do I have my story straight?

Also,

Question 1: $\tilde{X}$ is smooth and projective too, right?

Now for blow downs: I believe that in some special cases you can "blow down $\tilde{X}$ along $\tilde{Y}$ in the horizontal direction" to get another complex manifold $W$, different from $X$, and I think that this means that there is some complex submanifold $Z\subset W$ for which the blow up of $W$ along $Z$ is also equal to $\tilde{X}$ with the same exceptional divisor $\tilde{Y}$ realized now as the projectivized normal bundle of $Z\subset W$.

I guess I'm most used to thinking about polytopes and toric varieties. So for example take $\tilde{X}$ to be the toric variety corresponding to a truncated square pyramid, where moving one pair of opposite sides would result in one triangular prism (corresponding to a toric variety $X$), and moving the other pair would result in a different prism (corresponding to a different variety $W$). Here $Y\subset X$ and $Z\subset W$ would be the $\mathbb{CP}^1$'s corresponding to the top edge of their respective prisms and $\tilde{Y}$ would be the product $\mathbb{CP}^1\times\mathbb{CP}^1$.

Question 2: Are there general conditions on $Y\subset X$ that guarantee that we can blow up $X$ along $Y$, then blow down $\tilde{X}$ along $\tilde{Y}$ "in the horizontal direction" to get a new manifold $W$ as in the toric variety example?

From the example above, it seems like under such conditions we should always have:

1. $\tilde{Y}\cong\mathbb{CP}^{m-1}\times\mathbb{CP}^{n-1}$ (where $m$ is the codimension of $Z\subset W$),

2. and $H(\tilde{Y})\cong H(\mathbb{CP}^{m-1})\otimes_{\mathbb{R}} H(\mathbb{CP}^{n-1})\cong \mathbb{R}[x,y]/\langle x^m,y^n\rangle$ (where $x\in H^2(\mathbb{CP}^{m-1})$ and $y\in H^2(\mathbb{CP}^{n-1})$ are the hyperplane classes).

Question 3: In this case, can we explicitly identify the Chern class $z$ as an element of $\mathbb{R}[x,y]/\langle x^m,y^n\rangle$?

The condition that $z$ restricts to every fiber to give minus the hyperplane class seems to indicate that $z=-x-y$? Is that correct or am I missing something?

Finally I have one more general question: I think it's the case that every smooth toric variety can be obtained from a complex projective space by a finite sequence of blow ups and blow downs.

Question 4: What other smooth projective manifolds can be obtained from a projective space this way? Do such things have a name?

References and/or partial answers (and of course corrections) are welcome.

Answer 1

There are many questions here. Let me focus on Question 4.

The answer is all smooth rational varieties. This is a consequence of the following

Weak Factorization Theorem. A birational map between complete nonsingular varieties over an algebraically closed field $\mathbb{K}$ of characteristic zero is a composite of blowings up and blowings down with smooth centers.

References are:

1 D. Abramovich, K. Karu, K. Matsuki and J. Wlodarczyk: Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), 531-572

2 J. Wlodarczyk: Toroidal varieties and the weak factorization theorem, Invent. Math. 154, Issue 2 (2003), 223-331.