Let $X \subset \mathbb{P}^N$ be a nonsingular projective variety over algebraically closed field which is embedded by very ample line bundle $\mathcal{L}$. Let $Y = \mathbb{P}(\mathcal{L}^{\oplus 3})$ defined on $X$ and $\pi : Y \to X$ be the projection map. Let $U$ be an open subset of $X$ such that $\pi^{-1}(U)$ is isomorphic to $\mathbb{P}^2 \times U$. Let $t, s, r$ be coordinates of the fiber and $V$ be the subscheme of $\pi^{-1}(U)$ defined by $t + s + r = 0$. Let $D$ be the cycle of closure of $V$ in $Y$. Since $Y$ is nonsingular and the codimension of $V$ is 1, we can regard $D$ as a divisor of $Y$.
Let $\zeta$ be a first Chern class of tautological line bundle of $Y$. Is there any method to represent $D$ as a linear combination of $\zeta$ and $\pi^*\alpha$ for a cycle $\alpha$ of $X$?
If $U$ is a complement of a divisor of $X$, can we represent $D$ by $\zeta$ and $\pi^* \alpha$?
