In $\mathbb{P}^1\times\mathbb{P}^2$ take a general divisor $X$ of type $(0,2)$. Consider two general divisors $H\_1,H\_2$ of type $(2,1)$ and set $Y = X\cap H\_1\cap H\_2$.


Let $Z$ be the blow-up of $X$ along $Y$. I would like to ask whether there is a rank $3$ vector bundle $\mathbb{E}$ on $\mathbb{P}^1$ such that $Z$ can be embedded as a divisor in $\mathbb{P}(\mathbb{E})$ and if so how the splitting type of $\mathbb{E}$ can be computed.


Thank you very much.



Embedding of a blow-up
The map $Z \to \mathbb{P}^1$ is a conic bundle, so to understand the vector bundle $\mathbb{E}$ it is enough to compute the pushforward of the anticanonical class.


Now, the anticanonical class of $Z$ can be written as $-K\_X - E$, where $E$ is the exceptional divisor of the blowup, so its pushforward to $X$ is isomorphic to $I\_Y(-K\_X)$. If you use the identification of $X$ with $\mathbb{P}^1 \times \mathbb{P}^1$, this is $I\_Y(2,2)$. Using the Koszul resolution for $I\_Y$, one obtains
$$
0 \to \mathcal{O}(-2,-2) \to \mathcal{O}^{\oplus 2} \to I\_Y(2,2) \to 0.
$$
Pushing this forward to the first factor, one obtains
$$
0 \to \mathcal{O}^{\oplus 2} \to p\_{1\*}(I\_Y(2,2)) \to \mathcal{O}(-2) \to 0.
$$
Therefore, $\mathbb{E}$ has splitting type $(0,0,-2)$ or $(0,0,2)$, depending on which convention about the projective bundle you use.


