# embedding between projective bundles

Let $$X=\mathbb{P}_{\mathbb{P^1}}(\mathcal O(-1)+\mathcal O(-1)+\mathcal O)$$, $$Y=\mathbb{P}_{\mathbb{P^1}}(\mathcal O(-1)+\mathcal O+\mathcal O+\mathcal O)$$.

Can we embed $$X$$ into $$Y$$ as a hypersurface? If so, which divisor class of $$Y$$ would correspond to $$X$$?

Yes, there is an exact sequence $$0 \to \mathcal{O}(-1) \oplus \mathcal{O}(-1) \oplus \mathcal{O} \to \mathcal{O}(-1) \oplus \mathcal{O} \oplus \mathcal{O} \oplus \mathcal{O} \to \mathcal{O}(1) \to 0,$$ obtained as the direct sum of the standard exact sequence $$0 \to \mathcal{O}(-1) \to \mathcal{O} \oplus \mathcal{O} \to \mathcal{O}(1) \to 0$$ with $$\mathcal{O}(-1)$$ and $$\mathcal{O}$$. It induces the required embedding of projective bundles, and since the quotient is $$\mathcal{O}(1)$$, the image is linearly equivalent to $$H + h$$, where $$H$$ is the relative hyperplane class and $$h$$ is the pullback of the hyperplane class on $$\mathbb{P}^1$$.