Let $\mathbf{a}=(a\_0,\cdots,a\_n)\in\mathbb{Z}^{n+1}$ be an array of integers, let $\mathcal{E}(\mathbf{a})=\oplus\_{i=0}^n\mathcal{O}(a\_i)$ be the vector bundle on $\mathbb{P}^1$, and let $\mathbb{P}(\mathcal{E}(\mathbf{a}))$ be its projectivization. We know that $\mathbb{P}(\mathcal{E}(\mathbf{a}))$ and $\mathbb{P}(\mathcal{E}(\mathbf{b}))$ are isomorphic iff $\mathbf{a}-\mathbf{b}=(k,\cdots,k)$ for some $k\in\mathbb{Z}$.


For $n\geq2$, is there an explicit numerical criterion telling whether $\mathbb{P}(\mathcal{E}(\mathbf{a}))$ and $\mathbb{P}(\mathcal{E}(\mathbf{b}))$ are deformation equivalent?


(In case $n=1$, we know Hirzebruch surfaces are deformation equivalent iff $a\_1-a\_0\equiv b\_1-b\_0$ mod $2$)



Deformation equivalence of $\mathbb{P}^n$ bundles over $\mathbb{P}^1$
The projective bundles $\mathbb{P}(a)$ and $\mathbb{P}(b)$ are deformation equivalent if and only if
$$
\sum a\_i \equiv \sum b\_i \bmod n + 1.
$$
To show this it is enough to check that $\mathbb{P}(a)$ is deformation equivalent to $\mathbb{P}(a')$, where
$$
a' = (a\_0 - 1, a\_1 + 1, a\_2, \dots, a\_n).
$$
For this assume that $a\_0 \le a\_1$. Then there is an exact sequence
$$
0 \to \mathcal{O}(a\_0 - 1) \to \mathcal{O}(a\_0) \oplus \mathcal{O}(a\_1) \to \mathcal{O}(a\_1 + 1) \to 0,
$$
hence there is a deformation equivalence between vector bundles
$$
\mathcal{O}(a\_0 - 1) \oplus \mathcal{O}(a\_1 + 1)
\quad\text{and}\quad
\mathcal{O}(a\_0) \oplus \mathcal{O}(a\_1)
$$
that corresponds to deformation to zero of the extension class of the above exact sequence. Taking the sum with the other line bundles and passing to projectivizations, one obtains the required deformation equivalence for projective bundles.


